# A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Marc Munar, Miguel Couceiro, Sebastia Massanet, Daniel Ruiz-Aguilera

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Marc Munar, Miguel Couceiro, Sebastia Massanet, Daniel Ruiz-Aguilera. A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results. Fuzzy Sets and Systems, In press, 10.1016/j.fss.2024.109023. hal-04589444

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# A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Marc Munar<sup>a,b,*</sup>, Miguel Couceiro<sup>c,d</sup>, Sebastia Massanet<sup>a,b</sup> and Daniel Ruiz-Aguilera<sup>a,b</sup>

## ARTICLE INFO

**Keywords:**

logical connectives
aggregation functions
finite chains
enumeration
discrete t-norms
discrete implications

## ABSTRACT

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions.

## 1. Introduction

The exploration of logical connectives and aggregation functions defined on the finite chain $L_n = \{0, 1, ..., n\}$ has attracted the attention of the research community from its earliest stages. The finite nature of $L_n$ renders it a well-suited framework for modeling ordinal and qualitative information in different applied scenarios. Noteworthy instances include image processing employing discrete mathematical morphology [1, 2] and decision-making processes that hinge upon a predefined set of linguistic labels [3, 4]. In these contexts, the operators under consideration possess finite domain and image sets, so a finite number of operators can be constructed for each $n$. Consequently, when confronted with the task of determining the optimal configuration of these operators, one possible approach involves generating the complete set of operators, evaluating them, and subsequently selecting the one that best aligns with some criteria. However, this approach has inherent drawbacks. First, the enumeration of the desired class in this way may not be possible, as the cardinality of this class can grow rapidly, even for lower values of $n$, fact already realized by De Baets and Mesiar in [5] trying to generate all possible discrete t-norms for $n \le 13$, or Nachtegael and Kerre in [6] by computationally counting discrete implications and some subclasses. Second, in cases where comprehensive generation is feasible, forecasting the computational resources required for both operator generation and subsequent processing is essential. Third, building upon the previous points, it may not be feasible to generate the entire operator class due to its extensive cardinality, but it might be possible to generate specific subclasses with additional properties. Then, it becomes evident that having a prior formula for the number of operators on $L_n$ addresses several of these challenges. Moreover, this knowledge extends beyond mere enumeration, since it provides a deeper insight into the behavior of specific properties and how constraining the additional properties are in relation to its parent class.

<footnote><sup>a</sup>Soft Computing, Image Processing and Aggregation (SCOPIA) research group, Dept. of Mathematics and Computer Science, University of the Balearic Islands, Palma, 07122, Spain</footnote>

<footnote><sup>b</sup>Health Research Institute of the Balearic Islands (IdISBa), Palma, 07010, Spain</footnote>

<footnote><sup>c</sup>Université de Lorraine, CNRS, LORIA, Nancy, 54000, France</footnote>

<footnote><sup>d</sup>INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, 1000-029, Portugal</footnote>

<footnote>*Corresponding author</footnote>

<footnote>✉ marc.munar@uib.es (M. Munar); miguel.couceiro@loria.fr (M. Couceiro); s.massanet@uib.es (S. Massanet); daniel.ruiz@uib.es (D. Ruiz-Aguilera)</footnote>
# A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Marc Munar<sup>a,b,*</sup>, Miguel Couceiro<sup>c,d</sup>, Sebastia Massanet<sup>a,b</sup> and Daniel Ruiz-Aguilera<sup>a,b</sup>

## ARTICLE INFO

**Keywords:**

logical connectives
aggregation functions
finite chains
enumeration
discrete t-norms
discrete implications

## ABSTRACT

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions.

## 1. Introduction

The exploration of logical connectives and aggregation functions defined on the finite chain $L_n = \{0, 1, ..., n\}$ has attracted the attention of the research community from its earliest stages. The finite nature of $L_n$ renders it a well-suited framework for modeling ordinal and qualitative information in different applied scenarios. Noteworthy instances include image processing employing discrete mathematical morphology [1, 2] and decision-making processes that hinge upon a predefined set of linguistic labels [3, 4]. In these contexts, the operators under consideration possess finite domain and image sets, so a finite number of operators can be constructed for each $n$. Consequently, when confronted with the task of determining the optimal configuration of these operators, one possible approach involves generating the complete set of operators, evaluating them, and subsequently selecting the one that best aligns with some criteria. However, this approach has inherent drawbacks. First, the enumeration of the desired class in this way may not be possible, as the cardinality of this class can grow rapidly, even for lower values of $n$, fact already realized by De Baets and Mesiar in [5] trying to generate all possible discrete t-norms for $n \le 13$, or Nachtegael and Kerre in [6] by computationally counting discrete implications and some subclasses. Second, in cases where comprehensive generation is feasible, forecasting the computational resources required for both operator generation and subsequent processing is essential. Third, building upon the previous points, it may not be feasible to generate the entire operator class due to its extensive cardinality, but it might be possible to generate specific subclasses with additional properties. Then, it becomes evident that having a prior formula for the number of operators on $L_n$ addresses several of these challenges. Moreover, this knowledge extends beyond mere enumeration, since it provides a deeper insight into the behavior of specific properties and how constraining the additional properties are in relation to its parent class.

<footnote><sup>a</sup>Soft Computing, Image Processing and Aggregation (SCOPIA) research group, Dept. of Mathematics and Computer Science, University of the Balearic Islands, Palma, 07122, Spain</footnote>

<footnote><sup>b</sup>Health Research Institute of the Balearic Islands (IdISBa), Palma, 07010, Spain</footnote>

<footnote><sup>c</sup>Université de Lorraine, CNRS, LORIA, Nancy, 54000, France</footnote>

<footnote><sup>d</sup>INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, 1000-029, Portugal</footnote>

<footnote>*Corresponding author</footnote>

<footnote>✉ marc.munar@uib.es (M. Munar); miguel.couceiro@loria.fr (M. Couceiro); s.massanet@uib.es (S. Massanet); daniel.ruiz@uib.es (D. Ruiz-Aguilera)</footnote>
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Numerous papers in the literature have devoted their efforts to determine a closed formula for the enumeration of different classes of operators on $L_n$. In general terms, these enumerations have been approached from three distinct perspectives. The first perspective involves characterizing the class of operators, which serves to uncover its inherent structure, as exemplified in [7, 8, 9, 10, 11, 12]. This characterization often enables the establishment of relationships with other operators or structures for which enumeration formulas are previously known. The second perspective centers on the geometric attributes of the operator with some examples in [13, 14, 15, 16, 17]. This approach proves highly advantageous as it allows to relate theoretical properties with geometric characteristics and, in some instances, this relation simplifies the problem significantly. The third perspective revolves around establishing bijections between operators of $L_n$ and other mathematical objects, even if those objects are not intrinsically related to $L_n$, showing some cases of this strategy in [18, 19, 20, 21]. It is important to note that these three approaches are not mutually exclusive, and they are frequently employed in combination with each other.

This paper thus serves a dual purpose. Firstly, a comprehensive review of the existing literature on the enumeration of operators defined on $L_n$ and the results that lead to such enumerations is given, which can be considered as a continuation of [22], where a survey of operators on $L_n$ and relations between them is offered. In doing so, we consolidate all known formulas into a unified notation. Furthermore, since some of these formulas are originated from other frameworks, such as the study of semigroups, we translate these operators and their enumeration to $L_n$. In cases where the enumeration of a particular class of operators remains unknown, we take a computational approach to count them, providing some initial values of the enumerations for future reference. Additionally, we verify that these enumerations are appropriately cataloged in the Online Encyclopedia of Integer Sequences (OEIS) [23], the main database for integer sequences. Regarding the second objective, we introduce some novel formulas not previously documented. Namely, (i) the enumeration of smooth discrete aggregation functions that have neutral element or absorbing element, (ii) idempotent conjunctions, and (iii) commutative and idempotent conjunctions.

The paper is organized as follows. It is structured into distinct sections, each dedicated to the examination of a specific class of operators, presenting the corresponding operator definitions and enumerations. We categorize these operators into three main groups. The first group focuses on unary operators, specifically discrete negations, and they are explored in Section 2. The second group delves into discrete aggregation functions that are surveyed in Section 3. We take an in-depth look at various subclasses, including discrete conjunctions and disjunctions in Section 3.2, discrete quasi-copulas and copulas in Section 3.3, discrete t-norms and t-conorms in Section 3.4, discrete uninorms in Section 3.5, and discrete t-operators and nullnorms in Section 3.6. The third category encompasses discrete implications and some additional properties of them, which are discussed in Section 4. Subsequently, in Section 5, we introduce some new enumerations. Namely, formulas for the cardinality of smooth discrete aggregation functions, having neutral element or absorbing element, idempotent discrete conjunctions, and commutative and idempotent discrete conjunctions. The paper then concludes in Section 6, where we offer some final thoughts and provide a summary of the different enumerations along with their cataloging in the OEIS.

## 2. Enumeration of discrete negations

Discrete negations are unary operators that extend the negation $\neg$ in Boolean algebra. They possess undeniable utility, both on [0, 1] and on $L_n$. Their significance lies not only in their capacity to embody this broader generalization but also in their potential to generate new classes of operators from other already established.

**Definition 2.1.** ([8, 22]) A discrete negation is a unary operator $N : L_n \to L_n$ that satisfies the following properties:

$$ (N1) \quad N \text{ is decreasing; that is, } N(x) \geq N(y) \text{ for all } x, y \in L_n \text{ with } x \leq y. $$

$$ (N2) \quad N(0) = n \text{ and } N(n) = 0. $$

Apart from these two properties, additional properties of a discrete negation can be considered. One of them is that of being strong (or involutive), i.e., when $N(N(x)) = x$ holds for all $x \in L_n$. In discrete settings,
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Numerous papers in the literature have devoted their efforts to determine a closed formula for the enumeration of different classes of operators on $L_n$. In general terms, these enumerations have been approached from three distinct perspectives. The first perspective involves characterizing the class of operators, which serves to uncover its inherent structure, as exemplified in [7, 8, 9, 10, 11, 12]. This characterization often enables the establishment of relationships with other operators or structures for which enumeration formulas are previously known. The second perspective centers on the geometric attributes of the operator with some examples in [13, 14, 15, 16, 17]. This approach proves highly advantageous as it allows to relate theoretical properties with geometric characteristics and, in some instances, this relation simplifies the problem significantly. The third perspective revolves around establishing bijections between operators of $L_n$ and other mathematical objects, even if those objects are not intrinsically related to $L_n$, showing some cases of this strategy in [18, 19, 20, 21]. It is important to note that these three approaches are not mutually exclusive, and they are frequently employed in combination with each other.

This paper thus serves a dual purpose. Firstly, a comprehensive review of the existing literature on the enumeration of operators defined on $L_n$ and the results that lead to such enumerations is given, which can be considered as a continuation of [22], where a survey of operators on $L_n$ and relations between them is offered. In doing so, we consolidate all known formulas into a unified notation. Furthermore, since some of these formulas are originated from other frameworks, such as the study of semigroups, we translate these operators and their enumeration to $L_n$. In cases where the enumeration of a particular class of operators remains unknown, we take a computational approach to count them, providing some initial values of the enumerations for future reference. Additionally, we verify that these enumerations are appropriately cataloged in the Online Encyclopedia of Integer Sequences (OEIS) [23], the main database for integer sequences. Regarding the second objective, we introduce some novel formulas not previously documented. Namely, (i) the enumeration of smooth discrete aggregation functions that have neutral element or absorbing element, (ii) idempotent conjunctions, and (iii) commutative and idempotent conjunctions.

The paper is organized as follows. It is structured into distinct sections, each dedicated to the examination of a specific class of operators, presenting the corresponding operator definitions and enumerations. We categorize these operators into three main groups. The first group focuses on unary operators, specifically discrete negations, and they are explored in Section 2. The second group delves into discrete aggregation functions that are surveyed in Section 3. We take an in-depth look at various subclasses, including discrete conjunctions and disjunctions in Section 3.2, discrete quasi-copulas and copulas in Section 3.3, discrete t-norms and t-conorms in Section 3.4, discrete uninorms in Section 3.5, and discrete t-operators and nullnorms in Section 3.6. The third category encompasses discrete implications and some additional properties of them, which are discussed in Section 4. Subsequently, in Section 5, we introduce some new enumerations. Namely, formulas for the cardinality of smooth discrete aggregation functions, having neutral element or absorbing element, idempotent discrete conjunctions, and commutative and idempotent discrete conjunctions. The paper then concludes in Section 6, where we offer some final thoughts and provide a summary of the different enumerations along with their cataloging in the OEIS.

## 2. Enumeration of discrete negations

Discrete negations are unary operators that extend the negation $\neg$ in Boolean algebra. They possess undeniable utility, both on [0, 1] and on $L_n$. Their significance lies not only in their capacity to embody this broader generalization but also in their potential to generate new classes of operators from other already established.

**Definition 2.1.** ([8, 22]) A discrete negation is a unary operator $N : L_n \to L_n$ that satisfies the following properties:

$$ (N1) \quad N \text{ is decreasing; that is, } N(x) \ge N(y) \text{ for all } x, y \in L_n \text{ with } x \le y. $$

$$ (N2) \quad N(0) = n \text{ and } N(n) = 0. $$

Apart from these two properties, additional properties of a discrete negation can be considered. One of them is that of being strong (or involutive), i.e., when $N(N(x)) = x$ holds for all $x \in L_n$. In discrete settings,
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

it is straightforward to prove that a discrete negation is strong if, and only if, $N(x) = N_C(x) = n - x^1$ for all $x \in L_n$ [8], and therefore $N_C$ (the classical discrete negation) is the only decreasing automorphism on $L_n$.

When we consider operators defined on [0, 1], depending on the scenario, an interesting property to be considered is continuity, which is not available on $L_n$. However, several papers have coined the concept of smoothness on $L_n$ (see [8, 24, 25]) and they have considered it as the counterpart of continuity on [0, 1]. For unary operators, its definition is the following.

**Definition 2.2.** ([8]) A unary operator $f: L_n \to L_n$ is said to be k-smooth (or simply smooth, when $k=1$) if

$$|f(x+1) - f(x)| \le k, \qquad (1)$$

for all $x \in L_n \setminus \{n\}$.

Because of condition (N2), it is also trivial to show that a discrete negation $N$ is smooth if, and only if, $N(x) = N_C(x)$ for all $x \in L_n$.

The initial study that establishes the enumeration of discrete negations in general is attributed to Munar et al. in [19]. In this study, the authors proved the following formula grounded in elemental combinatorial arguments. To achieve it, they studied the number of monotone unary operators defined between two finite chains $L_{m_1}$ and $L_{m_2}$.

**Lemma 2.1.** ([19, Lemma 3.2]) Let $m_1, m_2 \in \mathbb{N}$ with $m_1, m_2 \ge 1$. Then, the number of monotone unary mappings $f: \{1, \dots, m_1\} \to \{1, \dots, m_2\}$ is

$$\binom{m_1 + m_2 - 1}{m_2 - 1}. \qquad (2)$$

**Remark 2.1.** In the initial version of Lemma 2.1 presented in [19, Lemma 3.2], the condition $m_2 \ge m_1$ is established. Nevertheless, it can be proved in a straightforward manner that this condition is redundant, and its removal does not alter the validity of the lemma.

Let $N_n$ be the set of all discrete negations defined on $L_n$. Using Lemma 2.1, the authors of [19] proposed a formula for $|N_n|$.

**Proposition 2.1.** ([19, Proposition 3.1]) Let $n \ge 1$ and let us denote by $N_n$ the set of discrete negations defined on $L_n$. Then,

$$|N_n| = \binom{2n-1}{n}. \qquad (3)$$

Mayor and Torrens [8] studied those unary operators $\delta: L_n \to L_n$, which are (i) monotone increasing, (ii) $\delta(x) \le x$ for all $x \in L_n$, and (iii) satisfy the boundary conditions $\delta(0) = 0$ and $\delta(n) = n$. They referred to this class of operators as *diagonal sections*, an appellation inspired by their exploration of diagonal sections of binary operators, and they identified in [8, Remark 7.3.5] a formula for the cardinality of the set of diagonal sections, for all $n \ge 1$.

Although not intrinsically linked to discrete negations, it is evident that the operator $N_\delta(x) = n - \delta(x)$, for all $x \in L_n$, is a discrete negation satisfying $N(x) \ge n - x = N_C(x)$, which up to our knowledge have not yet been studied in the literature, and we will refer to them as *over anti-diagonal discrete negations*. Therefore, a bijection between all diagonal sections and over anti-diagonal discrete negations can be established, preserving its cardinality.

**Proposition 2.2.** ([8, Remark 7.3.5]) Let $n \ge 1$ and let us denote by $N_n^{\text{OAD}}$ the set of over anti-diagonal discrete negations defined on $L_n$. Then,

$$|N_n^{\text{OAD}}| = \frac{1}{n+1} \binom{2n}{n}, \qquad (4)$$

which is the n-th Catalan number.

In Table 1 we illustrate for $n \le 13$ the values of $|N_n|$ and $|N_n^{\text{OAD}}|$. To conclude this section, recalling the properties of being smooth or involutive, if $N_n^{\text{SMT}}$ denotes the set of discrete negations that are smooth, and $N_n^{\text{INV}}$ denotes the set of discrete negations that are involutive, then $N_n^{\text{SMT}} = N_n^{\text{INV}} = \{N_C\}$ for all $n \ge 1$, so $|N_n^{\text{SMT}}| = |N_n^{\text{INV}}| = 1$, for all $n \ge 1$.

<footnote n="1">¹Without loss of generality, we may assume that $L_n$ is equipped with the usual arithmetic of the integer numbers.</footnote>
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

it is straightforward to prove that a discrete negation is strong if, and only if, $N(x) = N_C(x) = n - x^1$ for all $x \in L_n$ [8], and therefore $N_C$ (the classical discrete negation) is the only decreasing automorphism on $L_n$.

When we consider operators defined on [0, 1], depending on the scenario, an interesting property to be considered is continuity, which is not available on $L_n$. However, several papers have coined the concept of smoothness on $L_n$ (see [8, 24, 25]) and they have considered it as the counterpart of continuity on [0, 1]. For unary operators, its definition is the following.

**Definition 2.2.** ([8]) A unary operator $f: L_n \to L_n$ is said to be k-smooth (or simply smooth, when $k=1$) if

$$|f(x+1) - f(x)| \le k, \qquad (1)$$

for all $x \in L_n \setminus \{n\}$.

Because of condition (N2), it is also trivial to show that a discrete negation $N$ is smooth if, and only if, $N(x) = N_C(x)$ for all $x \in L_n$.

The initial study that establishes the enumeration of discrete negations in general is attributed to Munar et al. in [19]. In this study, the authors proved the following formula grounded in elemental combinatorial arguments. To achieve it, they studied the number of monotone unary operators defined between two finite chains $L_{m_1}$ and $L_{m_2}$.

**Lemma 2.1.** ([19, Lemma 3.2]) Let $m_1, m_2 \in \mathbb{N}$ with $m_1, m_2 \ge 1$. Then, the number of monotone unary mappings $f: \{1, \dots, m_1\} \to \{1, \dots, m_2\}$ is

$$\binom{m_1 + m_2 - 1}{m_2 - 1}. \qquad (2)$$

**Remark 2.1.** In the initial version of Lemma 2.1 presented in [19, Lemma 3.2], the condition $m_2 \ge m_1$ is established. Nevertheless, it can be proved in a straightforward manner that this condition is redundant, and its removal does not alter the validity of the lemma.

Let $N_n$ be the set of all discrete negations defined on $L_n$. Using Lemma 2.1, the authors of [19] proposed a formula for $|N_n|$.

**Proposition 2.1.** ([19, Proposition 3.1]) Let $n \ge 1$ and let us denote by $N_n$ the set of discrete negations defined on $L_n$. Then,

$$|N_n| = \binom{2n-1}{n}. \qquad (3)$$

Mayor and Torrens [8] studied those unary operators $\delta: L_n \to L_n$, which are (i) monotone increasing, (ii) $\delta(x) \le x$ for all $x \in L_n$, and (iii) satisfy the boundary conditions $\delta(0) = 0$ and $\delta(n) = n$. They referred to this class of operators as *diagonal sections*, an appellation inspired by their exploration of diagonal sections of binary operators, and they identified in [8, Remark 7.3.5] a formula for the cardinality of the set of diagonal sections, for all $n \ge 1$.

Although not intrinsically linked to discrete negations, it is evident that the operator $N_\delta(x) = n - \delta(x)$, for all $x \in L_n$, is a discrete negation satisfying $N(x) \ge n - x = N_C(x)$, which up to our knowledge have not yet been studied in the literature, and we will refer to them as *over anti-diagonal discrete negations*. Therefore, a bijection between all diagonal sections and over anti-diagonal discrete negations can be established, preserving its cardinality.

**Proposition 2.2.** ([8, Remark 7.3.5]) Let $n \ge 1$ and let us denote by $N_n^{\text{OAD}}$ the set of over anti-diagonal discrete negations defined on $L_n$. Then,

$$|N_n^{\text{OAD}}| = \frac{1}{n+1} \binom{2n}{n}, \qquad (4)$$

which is the n-th Catalan number.

In Table 1 we illustrate for $n \le 13$ the values of $|N_n|$ and $|N_n^{\text{OAD}}|$. To conclude this section, recalling the properties of being smooth or involutive, if $N_n^{\text{SMT}}$ denotes the set of discrete negations that are smooth, and $N_n^{\text{INV}}$ denotes the set of discrete negations that are involutive, then $N_n^{\text{SMT}} = N_n^{\text{INV}} = \{N_C\}$ for all $n \ge 1$, so $|N_n^{\text{SMT}}| = |N_n^{\text{INV}}| = 1$, for all $n \ge 1$.

<footnote n="1">¹Without loss of generality, we may assume that $L_n$ is equipped with the usual arithmetic of the integer numbers.</footnote>
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 1**</caption><thead><tr><th>n</th><th>N'<sub>n</sub></th><th>N'<sub>n</sub><sup>OAD</sup></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>3 = 3 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>10 = 1 &middot; 10<sup>1</sup></td><td>5 = 5 &middot; 10<sup>0</sup></td></tr><tr><td>4</td><td>35 = 3.5 &middot; 10<sup>1</sup></td><td>14 = 1.4 &middot; 10<sup>1</sup></td></tr><tr><td>5</td><td>126 = 1.26 &middot; 10<sup>2</sup></td><td>42 = 4.2 &middot; 10<sup>1</sup></td></tr><tr><td>6</td><td>462 = 4.62 &middot; 10<sup>2</sup></td><td>132 = 1.32 &middot; 10<sup>2</sup></td></tr><tr><td>7</td><td>1,716 = 1.716 &middot; 10<sup>3</sup></td><td>429 = 4.29 &middot; 10<sup>2</sup></td></tr><tr><td>8</td><td>6,435 = 6.435 &middot; 10<sup>3</sup></td><td>1,430 = 1.43 &middot; 10<sup>3</sup></td></tr><tr><td>9</td><td>24,310 = 2.431 &middot; 10<sup>4</sup></td><td>4,862 = 4.862 &middot; 10<sup>3</sup></td></tr><tr><td>10</td><td>92,378 = 9.2378 &middot; 10<sup>4</sup></td><td>16,796 = 1.6796 &middot; 10<sup>4</sup></td></tr><tr><td>11</td><td>≈ 3.5272 &middot; 10<sup>5</sup></td><td>58,786 = 5.8786 &middot; 10<sup>4</sup></td></tr><tr><td>12</td><td>≈ 1.3521 &middot; 10<sup>6</sup></td><td>≈ 2.0801 &middot; 10<sup>5</sup></td></tr><tr><td>13</td><td>≈ 5.2003 &middot; 10<sup>6</sup></td><td>≈ 7.429 &middot; 10<sup>5</sup></td></tr></tbody></table>

Enumeration of discrete negations, given by Proposition 2.1, and over anti-diagonal discrete negations, given by Proposition 2.2, for $n \le 13$.

## 3. Enumeration of discrete aggregation functions and some subclasses

In this section, we focus on the known formulas for the cardinality of some classes of discrete aggregation functions. In Section 3.1 we recall the concept of discrete aggregation function, its main properties and some existing enumerations. Then, in Sections 3.2, 3.3, 3.4, 3.5 and 3.6 we discuss enumerations of the main classes of discrete aggregation functions.

### 3.1. Discrete aggregation functions

We start by revisiting discrete aggregation functions. They are defined as monotone binary operations with two boundary conditions.

**Definition 3.1.** ([22, 26, 27]) A discrete aggregation function is a binary operator $F : L_n^2 \to L_n$ which satisfies the following properties:

(A1) $F$ is increasing in each argument, that is, $F(x, y) \le F(x', y')$, for all $x, y, x', y' \in L_n$ with $x \le x'$ and $y \le y'$.

(A2) $F(0, 0) = 0$ and $F(n, n) = n$.

Since this definition is sufficiently generic, some additional properties have been proposed in the literature, which are recalled in the next definition.

**Definition 3.2.** ([22, 28]) Let $F : L_n^2 \to L_n$ be a discrete aggregation function. Then, $F$ is said to be

(i) commutative if $F(x, y) = F(y, x)$, for all $x, y \in L_n$;

(ii) associative if $F(F(x, y), z)) = F(x, F(y, z))$, for all $x, y, z \in L_n$;

(iii) bisymmetric if $F(F(x, y), F(u, v)) = F(F(x, u), F(y, v))$, for all $x, y, u, v \in L_n$;

(iv) idempotent if $F(x, x) = x$, for all $x \in L_n$, and idempotent-free if $F(x, x) \ne x$ for all $x \in L_n \setminus \{0, n\}$;

(v) quasitrivial if $F(x, y) \in \{x, y\}$, for all $x, y \in L_n$.

Moreover, $F$ satisfies

(vi) the 1-Lipschitz condition if $F(z, y) - F(x, y) \le z - x$ for all $x, y, z \in L_n$ with $x \le z$;

(vii) the intermediate value theorem; that is, if $z = F(x, y)$ and $z' = F(x, y')$ with $z < z'$, then for all $z'' \in [z, z']$ there exists some $y'' \in [y, y']$ such that $z'' = F(x, y'')$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 1**</caption><thead><tr><th>n</th><th>N'<sub>n</sub></th><th>N'<sub>n</sub><sup>OAD</sup></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 2 · 10<sup>0</sup></td></tr><tr><td>2</td><td>3 = 3 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>3</td><td>10 = 1 · 10<sup>1</sup></td><td>5 = 5 · 10<sup>0</sup></td></tr><tr><td>4</td><td>35 = 3.5 · 10<sup>1</sup></td><td>14 = 1.4 · 10<sup>1</sup></td></tr><tr><td>5</td><td>126 = 1.26 · 10<sup>2</sup></td><td>42 = 4.2 · 10<sup>1</sup></td></tr><tr><td>6</td><td>462 = 4.62 · 10<sup>2</sup></td><td>132 = 1.32 · 10<sup>2</sup></td></tr><tr><td>7</td><td>1,716 = 1.716 · 10<sup>3</sup></td><td>429 = 4.29 · 10<sup>2</sup></td></tr><tr><td>8</td><td>6,435 = 6.435 · 10<sup>3</sup></td><td>1,430 = 1.43 · 10<sup>3</sup></td></tr><tr><td>9</td><td>24,310 = 2.431 · 10<sup>4</sup></td><td>4,862 = 4.862 · 10<sup>3</sup></td></tr><tr><td>10</td><td>92,378 = 9.2378 · 10<sup>4</sup></td><td>16,796 = 1.6796 · 10<sup>4</sup></td></tr><tr><td>11</td><td>≈ 3.5272 · 10<sup>5</sup></td><td>58,786 = 5.8786 · 10<sup>4</sup></td></tr><tr><td>12</td><td>≈ 1.3521 · 10<sup>6</sup></td><td>≈ 2.0801 · 10<sup>5</sup></td></tr><tr><td>13</td><td>≈ 5.2003 · 10<sup>6</sup></td><td>≈ 7.429 · 10<sup>5</sup></td></tr></tbody></table>

Enumeration of discrete negations, given by Proposition 2.1, and over anti-diagonal discrete negations, given by Proposition 2.2, for $n \le 13$.

### 3. Enumeration of discrete aggregation functions and some subclasses

In this section, we focus on the known formulas for the cardinality of some classes of discrete aggregation functions. In Section 3.1 we recall the concept of discrete aggregation function, its main properties and some existing enumerations. Then, in Sections 3.2, 3.3, 3.4, 3.5 and 3.6 we discuss enumerations of the main classes of discrete aggregation functions.

#### 3.1. Discrete aggregation functions

We start by revisiting discrete aggregation functions. They are defined as monotone binary operations with two boundary conditions.

**Definition 3.1.** ([22, 26, 27]) A discrete aggregation function is a binary operator $F : L_n^2 \to L_n$ which satisfies the following properties:

(A1) $F$ is increasing in each argument, that is, $F(x, y) \le F(x', y')$, for all $x, y, x', y' \in L_n$ with $x \le x'$ and $y \le y'$.

(A2) $F(0, 0) = 0$ and $F(n, n) = n$.

Since this definition is sufficiently generic, some additional properties have been proposed in the literature, which are recalled in the next definition.

**Definition 3.2.** ([22, 28]) Let $F : L_n^2 \to L_n$ be a discrete aggregation function. Then, $F$ is said to be

(i) commutative if $F(x, y) = F(y, x)$, for all $x, y \in L_n$;

(ii) associative if $F(F(x, y), z)) = F(x, F(y, z))$, for all $x, y, z \in L_n$;

(iii) bisymmetric if $F(F(x, y), F(u, v)) = F(F(x, u), F(y, v))$, for all $x, y, u, v \in L_n$;

(iv) idempotent if $F(x, x) = x$, for all $x \in L_n$, and idempotent-free if $F(x, x) \ne x$ for all $x \in L_n \setminus \{0, n\}$;

(v) quasitrivial if $F(x, y) \in \{x, y\}$, for all $x, y \in L_n$.

Moreover, $F$ satisfies

(vi) the 1-Lipschitz condition if $F(z, y) - F(x, y) \le z - x$ for all $x, y, z \in L_n$ with $x \le z$;

(vii) the intermediate value theorem; that is, if $z = F(x, y)$ and $z' = F(x, y')$ with $z < z'$, then for all $z'' \in [z, z']$ there exists some $y'' \in [y, y']$ such that $z'' = F(x, y'')$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Remark 3.1.** Of the different properties mentioned in Definition 3.2, there are different terminologies to refer to the quasitrivial property. While in this paper we will refer to it as quasitrivial, in the literature there are papers where it is referred to as internal, as in [29, 30], locally internal in [31] or conservative in [16].

Apart from these properties, the presence of a neutral element and an absorbing element (also known as annihilator) have also been considered. Then, given a binary operator $F : L_n^2 \to L_n$, an element $e \in L_n$ is said to be a *neutral element of F* if $F(x, e) = F(e, x) = x$, for all $x \in L_n$, and an element $a \in L_n$ is said to be an *absorbing element of F* if $F(x, a) = F(a, x) = a$, for all $x \in L_n$. With these characteristics, it is evident that a neutral and an absorbing element cannot be equal, but they can coexist in the same operator.

Before proceeding with the compilation of known enumerations, it is worth noting that another property of substantial significance for any binary operator, is the concept of smoothness. Much like the smoothness observed in unary operators, smoothness for binary operators is commonly regarded as the property analogous to the continuity of binary operators defined on [0, 1].

**Definition 3.3.** ([8, 22]) A binary operator $F : L_n^2 \to L_n$ is said to be *k-smooth* (or simply smooth when $k=1$) if it is *k-smooth* in each argument; that is, if

$$|F(x + 1, y) - F(x, y)| \le k,$$

for all $x \in L_n \setminus \{n\}$ and $y \in L_n$, and

$$|F(x, y + 1) - F(x, y)| \le k,$$

for all $x \in L_n$ and $y \in L_n \setminus \{n\}$.

A first milestone in obtaining a general formula for the number of discrete aggregation functions was proposed by Munar et al. in [19], where the formula they proposed is based on plane partitions. These objects, initially introduced by MacMahon in [32, 33] and later studied in depth by Stanley in [34], can be understood, intuitively, as a two-dimensional array of integers satisfying certain growth constraints. Their formal definition is the following.

**Definition 3.4.** ([32, 33, 34]) A plane partition $\pi$ with shape $\Lambda = (\lambda_1, \dots, \lambda_r)$ is a two-dimensional array of positive integers such that $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_r$ and represented by

$$\begin{array}{ccccccc}
\pi_{r,1} & \cdots & \pi_{r,\lambda_r} & & & & \\
\vdots & \ddots & & \ddots & & & \\
\pi_{2,1} & \pi_{2,2} & \cdots & \cdots & \pi_{2,\lambda_2} & & \\
\pi_{1,1} & \pi_{1,2} & \cdots & \cdots & \cdots & \pi_{1,\lambda_1}
\end{array}$$

where each row and column are decreasing; that is, $\pi_{i,j} \ge \pi_{i,j+1}$ and $\pi_{i,j} \ge \pi_{i+1,j}$, whenever both sides are defined.

The enumeration of this object, as it appears in Definition 3.4, is not possible. Although the set of points $(i, j)$ that index the plane partition is finite and the components $i$ and $j$ are bounded, it should be noted that the values that each $\pi_{i,j}$ can take need not be bounded. For this reason, the concept of $(a, b, c)$-cubic plane partition arises, as the one that satisfies that $r = a, \lambda_1 \le b$ and $1 \le \pi_{i,j} \le c$, for all $1 \le i \le a$ and $1 \le j \le \lambda_i$. MacMahon found a formula for the number of this class of plane partitions [32, 33], which is given by the triple product

$$\prod_{i=1}^{a} \prod_{j=1}^{b} \prod_{k=1}^{c} \frac{i+j+k-1}{i+j+k-2}. \qquad (5)$$

Let $A_n$ denote the set of discrete aggregation functions defined on $L_n$. In [19], the authors performed a decomposition of $A_n$ into several mutually disjoint subsets of operators, and established a bijection with different types of $(a, b, c)$-cubic plane partitions, allowing to establish the cardinality, as shown in the following result.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Remark 3.1.** Of the different properties mentioned in Definition 3.2, there are different terminologies to refer to the quasitrivial property. While in this paper we will refer to it as quasitrivial, in the literature there are papers where it is referred to as internal, as in [29, 30], locally internal in [31] or conservative in [16].

Apart from these properties, the presence of a neutral element and an absorbing element (also known as annihilator) have also been considered. Then, given a binary operator $F : L_n^2 \to L_n$, an element $e \in L_n$ is said to be a neutral element of $F$ if $F(x, e) = F(e, x) = x$, for all $x \in L_n$, and an element $a \in L_n$ is said to be an absorbing element of $F$ if $F(x, a) = F(a, x) = a$, for all $x \in L_n$. With these characteristics, it is evident that a neutral and an absorbing element cannot be equal, but they can coexist in the same operator.

Before proceeding with the compilation of known enumerations, it is worth noting that another property of substantial significance for any binary operator, is the concept of smoothness. Much like the smoothness observed in unary operators, smoothness for binary operators is commonly regarded as the property analogous to the continuity of binary operators defined on [0, 1].

**Definition 3.3.** ([8, 22]) A binary operator $F : L_n^2 \to L_n$ is said to be $k$-smooth (or simply smooth when $k = 1$) if it is $k$-smooth in each argument; that is, if

$$|F(x + 1, y) - F(x, y)| \le k,$$

for all $x \in L_n \setminus \{n\}$ and $y \in L_n$, and

$$|F(x, y + 1) - F(x, y)| \le k,$$

for all $x \in L_n$ and $y \in L_n \setminus \{n\}$.

A first milestone in obtaining a general formula for the number of discrete aggregation functions was proposed by Munar et al. in [19], where the formula they proposed is based on plane partitions. These objects, initially introduced by MacMahon in [32, 33] and later studied in depth by Stanley in [34], can be understood, intuitively, as a two-dimensional array of integers satisfying certain growth constraints. Their formal definition is the following.

**Definition 3.4.** ([32, 33, 34]) A plane partition $\pi$ with shape $\Lambda = (\lambda_1, \dots, \lambda_r)$ is a two-dimensional array of positive integers such that $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_r$ and represented by

$$\begin{array}{ccccccc}
\pi_{r,1} & \cdots & \pi_{r,\lambda_r} & & & & \\
\vdots & \ddots & & \ddots & & & \\
\pi_{2,1} & \pi_{2,2} & \cdots & \cdots & \pi_{2,\lambda_2} & & \\
\pi_{1,1} & \pi_{1,2} & \cdots & \cdots & \cdots & \pi_{1,\lambda_1}
\end{array}$$

where each row and column are decreasing; that is, $\pi_{i,j} \ge \pi_{i,j+1}$ and $\pi_{i,j} \ge \pi_{i+1,j}$, whenever both sides are defined.

The enumeration of this object, as it appears in Definition 3.4, is not possible. Although the set of points $(i, j)$ that index the plane partition is finite and the components $i$ and $j$ are bounded, it should be noted that the values that each $\pi_{i,j}$ can take need not be bounded. For this reason, the concept of $(a, b, c)$-cubic plane partition arises, as the one that satisfies that $r = a$, $\lambda_1 \le b$ and $1 \le \pi_{i,j} \le c$, for all $1 \le i \le a$ and $1 \le j \le \lambda_i$. MacMahon found a formula for the number of this class of plane partitions [32, 33], which is given by the triple product

$$\prod_{i=1}^{a} \prod_{j=1}^{b} \prod_{k=1}^{c} \frac{i+j+k-1}{i+j+k-2}. \qquad (5)$$

Let $A_n$ denote the set of discrete aggregation functions defined on $L_n$. In [19], the authors performed a decomposition of $A_n$ into several mutually disjoint subsets of operators, and established a bijection with different types of $(a, b, c)$-cubic plane partitions, allowing to establish the cardinality, as shown in the following result.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Proposition 3.1.** ([19, Proposition 4.4]) Let $n \ge 1$, and let us denote by $\mathcal{A}_n$ the set of discrete aggregation functions defined on $L_n$. Then,

$$|\mathcal{A}_n| = \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n} \frac{i+j+k-1}{i+j+k-2} - 2 \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n-1} \frac{i+j+k-1}{i+j+k-2} + \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n-2} \frac{i+j+k-1}{i+j+k-2}, \quad (6)$$

for all $n \ge 2$, and $|\mathcal{A}_1| = 4$.

**Remark 3.2.** Equation (6) comprises three nested products which are defined for $n \ge 2$. When $n = 2$, the final term in the sum entails a product in which the upper bound is less than the lower bound. In such instances, this product is conventionally treated as equivalent to 1.

After obtaining a general formula of $|\mathcal{A}_n|$, a particularly interesting line of research arises: the examination of the cardinality of various properties outlined in Definition 3.2. Regarding the commutativity, and building upon plane partitions, Munar et al. in [19] presented a formulation for determining the number of commutative discrete aggregation functions. This formulation employs the concept of symmetric plane partitions; that is, a plane partition $\pi$ satisfying $\pi_{i,j} = \pi_{j,i'}$, whenever both sides are defined. MacMahon in his seminal works [32, 33] conjectured a formula for the enumeration of symmetric $(a, a, c)$-cubic plane partitions, but it was Andrews in [35, 36] who gave a complete proof of the formula. This formula is given by

$$\prod_{i=1}^{a} \left( \frac{2i+c-1}{2i-1} \cdot \prod_{j=i+1}^{a} \frac{i+j+c-1}{i+j-1} \right). \quad (7)$$

With a similar strategy, the authors decomposed $|\mathcal{A}_n^{\text{COM}}|$ (the set of commutative discrete aggregation functions) into different mutually disjoint sets, and established bijections with some symmetric $(a, a, c)$-cubic plane partitions, leading to a general formula.

**Proposition 3.2.** ([19, Proposition 4.5]) Let $n \ge 1$, and let us denote by $\mathcal{A}_n^{\text{COM}}$ the set of commutative discrete aggregation functions defined on $L_n$. Then,

$$|\mathcal{A}_n^{\text{COM}}| = \prod_{i=1}^{n+1} \left( \frac{2i+n-1}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-1}{i+j-1} \right) - 2 \prod_{i=1}^{n+1} \left( \frac{2i+n-2}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-2}{i+j-1} \right) \\ + \prod_{i=1}^{n+1} \left( \frac{2i+n-3}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-3}{i+j-1} \right). \quad (8)$$

Now, let us direct our attention to the second property shown in Definition 3.2, namely, associativity. Before we proceed further, we want to note an observation. Typically, when formulating formulas for establishing the enumeration of a specific mathematical object, it is customary to translate its properties into geometric characteristics which tend to be more intuitive and, in the vast majority of cases, this enable an easier enumeration process. A paradigmatic illustration of this approach is the enumeration of plane partitions. Interestingly, certain properties lack a known geometric interpretation, associativity being one such example. In fact, a complete characterization of associativity in terms of properties that do not entail a functional equation remains an open challenge in the literature. Several studies have initiated investigations into the general characterization of associativity, as evidenced by some examples in [13, 17, 37, 38]. In [14, 16], it has been recognized that enumerating binary operators fulfilling associativity on its own presents challenges. Nonetheless, they highlight that in combination with other properties, it becomes feasible to characterize the class of operators and establish its enumeration.

We thus turn our attention to [16], where Couceiro et al. scrutinized jointly associativity and quasitriviality on discrete aggregation functions. It is important to emphasize that their approach diverged from the perspective of operators on $L_n$, instead focusing on binary operators defined on an arbitrary set $X$ (that can be a finite chain), which subsequently yield a semigroup (see [39] for more information about this structure). They establish equivalences between associativity and quasitriviality properties of
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Proposition 3.1.** ([19, Proposition 4.4]) Let $n \ge 1$, and let us denote by $\mathcal{A}_n$ the set of discrete aggregation functions defined on $L_n$. Then,

$$|\mathcal{A}_n| = \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n} \frac{i+j+k-1}{i+j+k-2} - 2 \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n-1} \frac{i+j+k-1}{i+j+k-2} + \prod_{i=1}^{n+1} \prod_{j=1}^{n+1} \prod_{k=1}^{n-2} \frac{i+j+k-1}{i+j+k-2}, \quad (6)$$

for all $n \ge 2$, and $|\mathcal{A}_1| = 4$.

**Remark 3.2.** Equation (6) comprises three nested products which are defined for $n \ge 2$. When $n = 2$, the final term in the sum entails a product in which the upper bound is less than the lower bound. In such instances, this product is conventionally treated as equivalent to 1.

After obtaining a general formula of $|\mathcal{A}_n|$, a particularly interesting line of research arises: the examination of the cardinality of various properties outlined in Definition 3.2. Regarding the commutativity, and building upon plane partitions, Munar et al. in [19] presented a formulation for determining the number of commutative discrete aggregation functions. This formulation employs the concept of symmetric plane partitions; that is, a plane partition $\pi$ satisfying $\pi_{i,j} = \pi_{j,i'}$, whenever both sides are defined. MacMahon in his seminal works [32, 33] conjectured a formula for the enumeration of symmetric $(a, a, c)$-cubic plane partitions, but it was Andrews in [35, 36] who gave a complete proof of the formula. This formula is given by

$$\prod_{i=1}^{a} \left( \frac{2i+c-1}{2i-1} \cdot \prod_{j=i+1}^{a} \frac{i+j+c-1}{i+j-1} \right). \quad (7)$$

With a similar strategy, the authors decomposed $|\mathcal{A}_n^{\text{COM}}|$ (the set of commutative discrete aggregation functions) into different mutually disjoint sets, and established bijections with some symmetric $(a, a, c)$-cubic plane partitions, leading to a general formula.

**Proposition 3.2.** ([19, Proposition 4.5]) Let $n \ge 1$, and let us denote by $\mathcal{A}_n^{\text{COM}}$ the set of commutative discrete aggregation functions defined on $L_n$. Then,

$$|\mathcal{A}_n^{\text{COM}}| = \prod_{i=1}^{n+1} \left( \frac{2i+n-1}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-1}{i+j-1} \right) - 2 \prod_{i=1}^{n+1} \left( \frac{2i+n-2}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-2}{i+j-1} \right) \\ + \prod_{i=1}^{n+1} \left( \frac{2i+n-3}{2i-1} \cdot \prod_{j=i+1}^{n+1} \frac{i+j+n-3}{i+j-1} \right). \quad (8)$$

Now, let us direct our attention to the second property shown in Definition 3.2, namely, associativity. Before we proceed further, we want to note an observation. Typically, when formulating formulas for establishing the enumeration of a specific mathematical object, it is customary to translate its properties into geometric characteristics which tend to be more intuitive and, in the vast majority of cases, this enable an easier enumeration process. A paradigmatic illustration of this approach is the enumeration of plane partitions. Interestingly, certain properties lack a known geometric interpretation, associativity being one such example. In fact, a complete characterization of associativity in terms of properties that do not entail a functional equation remains an open challenge in the literature. Several studies have initiated investigations into the general characterization of associativity, as evidenced by some examples in [13, 17, 37, 38]. In [14, 16], it has been recognized that enumerating binary operators fulfilling associativity on its own presents challenges. Nonetheless, they highlight that in combination with other properties, it becomes feasible to characterize the class of operators and establish its enumeration.

We thus turn our attention to [16], where Couceiro et al. scrutinized jointly associativity and quasitriviality on discrete aggregation functions. It is important to emphasize that their approach diverged from the perspective of operators on $L_n$, instead focusing on binary operators defined on an arbitrary set $X$ (that can be a finite chain), which subsequently yield a semigroup (see [39] for more information about this structure). They establish equivalences between associativity and quasitriviality properties of
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

a binary operator $F: X_n^2 \to X_n$ (which is not necessarily monotone) defined on $X_n = \{1, 2, ..., n\}$, and the representation of $F$ concerning weak orderings induced by $F$, as proved in [16, Theorem 2.1]. When they impose conditions related to monotonicity, these operators become associative and quasitrivial discrete aggregation functions on $L_n$ (which is isomorphic to $X_{n+1}$) since quasitriviality ensures that $F(0, 0) = 0$ and $F(n, n) = n$, and they are able to characterize these operators using the framework of weak orderings, as outlined in [16, Theorem 4.5]. These orderings establish connections with the contour lines of the operators and, hence, with their geometry. A similar process was carried out by Kiss in [17] which, chronologically, is simultaneous to [16]. He also characterized these operators, but from a point of view that only involves contour lines. Subsequently, through the enumeration of weak orderings and contour lines, they are able to determine the cardinality of associative and quasitrivial discrete aggregation functions, and some subclasses.

**Proposition 3.3.** Let $n \ge 1$, and let us denote by

* $\mathcal{A}_n^{\text{ASV}}$, the set of associative discrete aggregation functions defined on $L_n$;

* $\mathcal{A}_n^{\text{QST}}$, the set of quasitrivial discrete aggregation functions defined on $L_n$;

* $\mathcal{A}_n^{\text{NE}}$, the set of discrete aggregation functions defined on $L_n$ having some neutral element;

* $\mathcal{A}_n^{\text{ABS}}$, the set of discrete aggregation functions defined on $L_n$ having some absorbing element.

Then, the following statements hold:

(i) ([16, Proposition 4.11], [17, Theorem 6.1]) The number of associative and quasitrivial discrete aggregation functions, $|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}}|$, satisfies the recurrence equation

$$|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}}| - 2 \cdot |\mathcal{A}_{n-1}^{\text{ASV}} \cap \mathcal{A}_{n-1}^{\text{QST}}| - 2 \cdot |\mathcal{A}_{n-2}^{\text{ASV}} \cap \mathcal{A}_{n-2}^{\text{QST}}| = 2, \quad (9)$$

for all $n \ge 3$, with $|\mathcal{A}_1^{\text{ASV}} \cap \mathcal{A}_1^{\text{QST}}| = 4$ and $|\mathcal{A}_2^{\text{ASV}} \cap \mathcal{A}_2^{\text{QST}}| = 12$. Therefore, its general formula is given by

$$\begin{aligned} |\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}}| &= \sum_{k=0}^{\lfloor n+1/2 \rfloor} 2 \cdot 3^{k-1} \cdot \binom{n+1}{2k} + \sum_{k=0}^{\lfloor n/2 \rfloor} 3^k \cdot \binom{n+1}{2k+1} - \frac{2}{3} \\ &= \frac{2+\sqrt{3}}{6} (1+\sqrt{3})^{n+1} + \frac{2-\sqrt{3}}{6} (1-\sqrt{3})^{n+1} - \frac{2}{3}. \end{aligned} \quad (10)$$

(ii) ([16, Proposition 4.12], [17, Theorem 6.1]) The number of associative and quasitrivial discrete aggregation functions with neutral element, $|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}} \cap \mathcal{A}_n^{\text{NE}}|$, satisfies the recurrence equation

$$|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}} \cap \mathcal{A}_n^{\text{NE}}| - 2 \cdot |\mathcal{A}_{n-1}^{\text{ASV}} \cap \mathcal{A}_{n-1}^{\text{QST}} \cap \mathcal{A}_{n-1}^{\text{NE}}| - 2 \cdot |\mathcal{A}_{n-2}^{\text{ASV}} \cap \mathcal{A}_{n-2}^{\text{QST}} \cap \mathcal{A}_{n-2}^{\text{NE}}| = 0, \quad (11)$$

for all $n \ge 3$, with $|\mathcal{A}_1^{\text{ASV}} \cap \mathcal{A}_1^{\text{QST}} \cap \mathcal{A}_1^{\text{NE}}| = 2$ and $|\mathcal{A}_2^{\text{ASV}} \cap \mathcal{A}_2^{\text{QST}} \cap \mathcal{A}_2^{\text{NE}}| = 6$. Therefore, its general formula is given by

$$\begin{aligned} |\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}} \cap \mathcal{A}_n^{\text{NE}}| &= \sum_{k=0}^{\lfloor n/2 \rfloor} 3^k \cdot \binom{n+1}{2k+1} \\ &= \frac{\sqrt{3}}{6} (1+\sqrt{3})^{n+1} - \frac{\sqrt{3}}{6} (1-\sqrt{3})^{n+1}. \end{aligned} \quad (12)$$

(iii) ([16, Proposition 4.12]) The number of associative and quasitrivial discrete aggregation functions with absorbing element, $|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}} \cap \mathcal{A}_n^{\text{ABS}}|$, satisfies the recurrence relation

$$|\mathcal{A}_n^{\text{ASV}} \cap \mathcal{A}_n^{\text{QST}} \cap \mathcal{A}_n^{\text{ABS}}| = 2 \cdot |\mathcal{A}_{n-1}^{\text{ASV}} \cap \mathcal{A}_{n-1}^{\text{QST}}|, \quad (13)$$
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

a binary operator $F: X_n^2 \to X_n$ (which is not necessarily monotone) defined on $X_n = \{1, 2, \dots, n\}$, and the representation of $F$ concerning weak orderings induced by $F$, as proved in [16, Theorem 2.1]. When they impose conditions related to monotonicity, these operators become associative and quasitrivial discrete aggregation functions on $L_n$ (which is isomorphic to $X_{n+1}$) since quasitriviality ensures that $F(0, 0) = 0$ and $F(n, n) = n$, and they are able to characterize these operators using the framework of weak orderings, as outlined in [16, Theorem 4.5]. These orderings establish connections with the contour lines of the operators and, hence, with their geometry. A similar process was carried out by Kiss in [17] which, chronologically, is simultaneous to [16]. He also characterized these operators, but from a point of view that only involves contour lines. Subsequently, through the enumeration of weak orderings and contour lines, they are able to determine the cardinality of associative and quasitrivial discrete aggregation functions, and some subclasses.

**Proposition 3.3.** Let $n \ge 1$, and let us denote by

* $A_n^{\text{ASV}}$, the set of associative discrete aggregation functions defined on $L_n$;

* $A_n^{\text{QST}}$, the set of quasitrivial discrete aggregation functions defined on $L_n$;

* $A_n^{\text{NE}}$, the set of discrete aggregation functions defined on $L_n$ having some neutral element;

* $A_n^{\text{ABS}}$, the set of discrete aggregation functions defined on $L_n$ having some absorbing element.

Then, the following statements hold:

(i) ([16, Proposition 4.11], [17, Theorem 6.1]) The number of associative and quasitrivial discrete aggregation functions, $|A_n^{\text{ASV}} \cap A_n^{\text{QST}}|$, satisfies the recurrence equation

$$|A_n^{\text{ASV}} \cap A_n^{\text{QST}}| - 2 \cdot |A_{n-1}^{\text{ASV}} \cap A_{n-1}^{\text{QST}}| - 2 \cdot |A_{n-2}^{\text{ASV}} \cap A_{n-2}^{\text{QST}}| = 2, \quad (9)$$

for all $n \ge 3$, with $|A_1^{\text{ASV}} \cap A_1^{\text{QST}}| = 4$ and $|A_2^{\text{ASV}} \cap A_2^{\text{QST}}| = 12$. Therefore, its general formula is given by

$$\begin{aligned} |A_n^{\text{ASV}} \cap A_n^{\text{QST}}| &= \sum_{k=0}^{\lfloor n+1/2 \rfloor} 2 \cdot 3^{k-1} \cdot \binom{n+1}{2k} + \sum_{k=0}^{\lfloor n/2 \rfloor} 3^k \cdot \binom{n+1}{2k+1} - \frac{2}{3} \\ &= \frac{2+\sqrt{3}}{6} (1+\sqrt{3})^{n+1} + \frac{2-\sqrt{3}}{6} (1-\sqrt{3})^{n+1} - \frac{2}{3}. \end{aligned} \quad (10)$$

(ii) ([16, Proposition 4.12], [17, Theorem 6.1]) The number of associative and quasitrivial discrete aggregation functions with neutral element, $|A_n^{\text{ASV}} \cap A_n^{\text{QST}} \cap A_n^{\text{NE}}|$, satisfies the recurrence equation

$$|A_n^{\text{ASV}} \cap A_n^{\text{QST}} \cap A_n^{\text{NE}}| - 2 \cdot |A_{n-1}^{\text{ASV}} \cap A_{n-1}^{\text{QST}} \cap A_{n-1}^{\text{NE}}| - 2 \cdot |A_{n-2}^{\text{ASV}} \cap A_{n-2}^{\text{QST}} \cap A_{n-2}^{\text{NE}}| = 0, \quad (11)$$

for all $n \ge 3$, with $|A_1^{\text{ASV}} \cap A_1^{\text{QST}} \cap A_1^{\text{NE}}| = 2$ and $|A_2^{\text{ASV}} \cap A_2^{\text{QST}} \cap A_2^{\text{NE}}| = 6$. Therefore, its general formula is given by

$$\begin{aligned} |A_n^{\text{ASV}} \cap A_n^{\text{QST}} \cap A_n^{\text{NE}}| &= \sum_{k=0}^{\lfloor n/2 \rfloor} 3^k \cdot \binom{n+1}{2k+1} \\ &= \frac{\sqrt{3}}{6} (1+\sqrt{3})^{n+1} - \frac{\sqrt{3}}{6} (1-\sqrt{3})^{n+1}. \end{aligned} \quad (12)$$

(iii) ([16, Proposition 4.12]) The number of associative and quasitrivial discrete aggregation functions with absorbing element, $|A_n^{\text{ASV}} \cap A_n^{\text{QST}} \cap A_n^{\text{ABS}}|$, satisfies the recurrence relation

$$|A_n^{\text{ASV}} \cap A_n^{\text{QST}} \cap A_n^{\text{ABS}}| = 2 \cdot |A_{n-1}^{\text{ASV}} \cap A_{n-1}^{\text{QST}}|, \quad (13)$$
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><thead><tr><th>n</th><th>|A<sub>n</sub>|</th><th>|A<sub>n</sub><sup>COM</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th></tr></thead><tbody><tr><td>1</td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 1 · 10<sup>0</sup></td></tr><tr><td>2</td><td>136 = 1.36 · 10<sup>2</sup></td><td>20 = 2 · 10<sup>1</sup></td><td>10 = 1 · 10<sup>1</sup></td><td>4 = 4 · 10<sup>0</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>3</td><td>21,238 = 2.1238 · 10<sup>4</sup></td><td>436 = 4.36 · 10<sup>2</sup></td><td>22 = 2.2 · 10<sup>1</sup></td><td>8 = 8 · 10<sup>0</sup></td><td>20 = 2.05 · 10<sup>1</sup></td></tr><tr><td>4</td><td>≈ 1.5374 · 10<sup>7</sup></td><td>20,328 = 2.0328 · 10<sup>4</sup></td><td>46 = 4.6 · 10<sup>1</sup></td><td>16 = 1.6 · 10<sup>1</sup></td><td>44 = 4.4 · 10<sup>1</sup></td></tr><tr><td>5</td><td>≈ 5.2327 · 10<sup>10</sup></td><td>≈ 2.0315 · 10<sup>6</sup></td><td>94 = 9.4 · 10<sup>1</sup></td><td>32 = 3.2 · 10<sup>1</sup></td><td>92 = 9.2 · 10<sup>1</sup></td></tr><tr><td>6</td><td>≈ 8.4502 · 10<sup>14</sup></td><td>≈ 4.372 · 10<sup>8</sup></td><td>190 = 1.9 · 10<sup>2</sup></td><td>64 = 6.4 · 10<sup>1</sup></td><td>188 = 1.88 · 10<sup>2</sup></td></tr><tr><td>7</td><td>≈ 6.5103 · 10<sup>19</sup></td><td>≈ 2.0348 · 10<sup>11</sup></td><td>382 = 3.82 · 10<sup>2</sup></td><td>128 = 1.28 · 10<sup>2</sup></td><td>380 = 3.8 · 10<sup>2</sup></td></tr><tr><td>8</td><td>≈ 2.4006 · 10<sup>25</sup></td><td>≈ 2.0548 · 10<sup>14</sup></td><td>766 = 7.66 · 10<sup>2</sup></td><td>256 = 2.56 · 10<sup>2</sup></td><td>764 = 7.64 · 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 4.2446 · 10<sup>31</sup></td><td>≈ 4.5137 · 10<sup>17</sup></td><td>1,534 = 1.534 · 10<sup>3</sup></td><td>512 = 5.12 · 10<sup>2</sup></td><td>1,532 = 1.532 · 10<sup>3</sup></td></tr><tr><td>10</td><td>≈ 3.6026 · 10<sup>38</sup></td><td>≈ 2.1607 · 10<sup>21</sup></td><td>3,070 = 3.07 · 10<sup>3</sup></td><td>1,024 = 1.024 · 10<sup>3</sup></td><td>3,068 = 3.068 · 10<sup>3</sup></td></tr><tr><td>11</td><td>≈ 1.4687 · 10<sup>46</sup></td><td>≈ 2.2573 · 10<sup>25</sup></td><td>6,142 = 6.142 · 10<sup>3</sup></td><td>2,048 = 2.048 · 10<sup>3</sup></td><td>6,140 = 6.14 · 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 2.8769 · 10<sup>54</sup></td><td>≈ 5.1522 · 10<sup>29</sup></td><td>12,286 = 1.2286 · 10<sup>4</sup></td><td>4,096 = 4.096 · 10<sup>3</sup></td><td>12,284 = 1.2284 · 10<sup>4</sup></td></tr><tr><td>13</td><td>≈ 2.7083 · 10<sup>63</sup></td><td>≈ 2.5713 · 10<sup>34</sup></td><td>24,574 = 2.4574 · 10<sup>4</sup></td><td>8,192 = 8.192 · 10<sup>3</sup></td><td>24,572 = 2.4572 · 10<sup>4</sup></td></tr></tbody></table>

<table><caption>Table 2</caption><thead><tr><th>n</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th></tr></thead><tbody><tr><td>1</td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>2</td><td>12 = 1.2 · 10<sup>1</sup></td><td>6 = 6 · 10<sup>0</sup></td><td>8 = 8 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td></tr><tr><td>3</td><td>34 = 3.4 · 10<sup>1</sup></td><td>16 = 1.6 · 10<sup>1</sup></td><td>24 = 2.4 · 10<sup>1</sup></td><td>12 = 1.2 · 10<sup>1</sup></td></tr><tr><td>4</td><td>94 = 9.4 · 10<sup>1</sup></td><td>44 = 4.4 · 10<sup>1</sup></td><td>68 = 6.8 · 10<sup>1</sup></td><td>32 = 3.2 · 10<sup>1</sup></td></tr><tr><td>5</td><td>258 = 2.58 · 10<sup>2</sup></td><td>120 = 1.2 · 10<sup>2</sup></td><td>188 = 1.88 · 10<sup>2</sup></td><td>88 = 8.8 · 10<sup>1</sup></td></tr><tr><td>6</td><td>706 = 7.06 · 10<sup>2</sup></td><td>328 = 3.28 · 10<sup>2</sup></td><td>516 = 5.16 · 10<sup>2</sup></td><td>240 = 2.4 · 10<sup>2</sup></td></tr><tr><td>7</td><td>1,930 = 1.93 · 10<sup>3</sup></td><td>896 = 8.96 · 10<sup>2</sup></td><td>1,412 = 1.412 · 10<sup>3</sup></td><td>656 = 6.56 · 10<sup>2</sup></td></tr><tr><td>8</td><td>5,274 = 5.274 · 10<sup>3</sup></td><td>2,448 = 2.448 · 10<sup>3</sup></td><td>3,860 = 3.86 · 10<sup>3</sup></td><td>1,792 = 1.792 · 10<sup>3</sup></td></tr><tr><td>9</td><td>14,410 = 1.441 · 10<sup>4</sup></td><td>6,688 = 6.688 · 10<sup>3</sup></td><td>10,548 = 1.0548 · 10<sup>4</sup></td><td>4,896 = 4.896 · 10<sup>3</sup></td></tr><tr><td>10</td><td>39,370 = 3.937 · 10<sup>4</sup></td><td>18,272 = 1.8272 · 10<sup>4</sup></td><td>28,820 = 2.882 · 10<sup>4</sup></td><td>13,376 = 1.3376 · 10<sup>4</sup></td></tr><tr><td>11</td><td>≈ 1.0756 · 10<sup>5</sup></td><td>49,920 = 4.992 · 10<sup>4</sup></td><td>78,740 = 7.874 · 10<sup>4</sup></td><td>36,544 = 3.6544 · 10<sup>4</sup></td></tr><tr><td>12</td><td>≈ 2.9387 · 10<sup>5</sup></td><td>≈ 1.3638 · 10<sup>5</sup></td><td>≈ 2.1512 · 10<sup>5</sup></td><td>99,840 = 9.984 · 10<sup>4</sup></td></tr><tr><td>13</td><td>≈ 8.0286 · 10<sup>5</sup></td><td>≈ 3.7261 · 10<sup>5</sup></td><td>≈ 5.8773 · 10<sup>5</sup></td><td>≈ 2.7277 · 10<sup>5</sup></td></tr></tbody></table>

Enumeration of some classes of discrete aggregation functions for $n \le 13$. From top to bottom, and left to right: discrete aggregation functions in general, given by Proposition 3.1; commutative discrete aggregation functions, given by Proposition 3.2; bisymmetric and quasitrivial discrete aggregation functions, bisymmetric and quasitrivial discrete aggregation functions with neutral element and bisymmetric and quasitrivial discrete aggregation functions with absorbing element, given by Proposition 3.4; and finally associative and quasitrivial discrete aggregation functions, associative and quasitrivial discrete aggregation functions with neutral element, associative and quasitrivial discrete aggregation functions with absorbing element, and associative and quasitrivial discrete aggregation functions with neutral and absorbing elements, given by Proposition 3.3.

(iii) ([28, Proposition 4.6]) The number of bisymmetric and quasitrivial discrete aggregation functions with absorbing element, $|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}|$, satisfies the recurrence equation

$$|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}| = |A_n^{BSM} \cap A_n^{QST}| - 2, \qquad (20)$$

for all $n \ge 1$, with $|A_1^{BSM} \cap A_1^{QST} \cap A_1^{ABS}| = 2$. Therefore, its general formula is given by

$$|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}| = 3 \cdot 2^n - 4. \qquad (21)$$

In Table 2 we list the first 13 values of all the formulas that have appeared in this section.

## 3.2. Discrete conjunctions and disjunctions

As noteworthy examples of discrete aggregation functions, we have the so-called discrete conjunctions and discrete disjunctions.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><thead><tr><th>n</th><th>|A<sub>n</sub>|</th><th>|A<sub>n</sub><sup>COM</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>|A<sub>n</sub><sup>BSM</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th></tr></thead><tbody><tr><td>1</td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 1 · 10<sup>0</sup></td></tr><tr><td>2</td><td>136 = 1.36 · 10<sup>2</sup></td><td>20 = 2 · 10<sup>1</sup></td><td>10 = 1 · 10<sup>1</sup></td><td>4 = 4 · 10<sup>0</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>3</td><td>21,238 = 2.1238 · 10<sup>4</sup></td><td>436 = 4.36 · 10<sup>2</sup></td><td>22 = 2.2 · 10<sup>1</sup></td><td>8 = 8 · 10<sup>0</sup></td><td>20 = 2.05 · 10<sup>1</sup></td></tr><tr><td>4</td><td>≈ 1.5374 · 10<sup>7</sup></td><td>20,328 = 2.0328 · 10<sup>4</sup></td><td>46 = 4.6 · 10<sup>1</sup></td><td>16 = 1.6 · 10<sup>1</sup></td><td>44 = 4.4 · 10<sup>1</sup></td></tr><tr><td>5</td><td>≈ 5.2327 · 10<sup>10</sup></td><td>≈ 2.0315 · 10<sup>6</sup></td><td>94 = 9.4 · 10<sup>1</sup></td><td>32 = 3.2 · 10<sup>1</sup></td><td>92 = 9.2 · 10<sup>1</sup></td></tr><tr><td>6</td><td>≈ 8.4502 · 10<sup>14</sup></td><td>≈ 4.372 · 10<sup>8</sup></td><td>190 = 1.9 · 10<sup>2</sup></td><td>64 = 6.4 · 10<sup>1</sup></td><td>188 = 1.88 · 10<sup>2</sup></td></tr><tr><td>7</td><td>≈ 6.5103 · 10<sup>19</sup></td><td>≈ 2.0348 · 10<sup>11</sup></td><td>382 = 3.82 · 10<sup>2</sup></td><td>128 = 1.28 · 10<sup>2</sup></td><td>380 = 3.8 · 10<sup>2</sup></td></tr><tr><td>8</td><td>≈ 2.4006 · 10<sup>25</sup></td><td>≈ 2.0548 · 10<sup>14</sup></td><td>766 = 7.66 · 10<sup>2</sup></td><td>256 = 2.56 · 10<sup>2</sup></td><td>764 = 7.64 · 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 4.2446 · 10<sup>31</sup></td><td>≈ 4.5137 · 10<sup>17</sup></td><td>1,534 = 1.534 · 10<sup>3</sup></td><td>512 = 5.12 · 10<sup>2</sup></td><td>1,532 = 1.532 · 10<sup>3</sup></td></tr><tr><td>10</td><td>≈ 3.6026 · 10<sup>38</sup></td><td>≈ 2.1607 · 10<sup>21</sup></td><td>3,070 = 3.07 · 10<sup>3</sup></td><td>1,024 = 1.024 · 10<sup>3</sup></td><td>3,068 = 3.068 · 10<sup>3</sup></td></tr><tr><td>11</td><td>≈ 1.4687 · 10<sup>46</sup></td><td>≈ 2.2573 · 10<sup>25</sup></td><td>6,142 = 6.142 · 10<sup>3</sup></td><td>2,048 = 2.048 · 10<sup>3</sup></td><td>6,140 = 6.14 · 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 2.8769 · 10<sup>54</sup></td><td>≈ 5.1522 · 10<sup>29</sup></td><td>12,286 = 1.2286 · 10<sup>4</sup></td><td>4,096 = 4.096 · 10<sup>3</sup></td><td>12,284 = 1.2284 · 10<sup>4</sup></td></tr><tr><td>13</td><td>≈ 2.7083 · 10<sup>63</sup></td><td>≈ 2.5713 · 10<sup>34</sup></td><td>24,574 = 2.4574 · 10<sup>4</sup></td><td>8,192 = 8.192 · 10<sup>3</sup></td><td>24,572 = 2.4572 · 10<sup>4</sup></td></tr></tbody></table>

<table><caption>Table 2</caption><thead><tr><th>n</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th><th>|A<sub>n</sub><sup>ASV</sup> ∩ A<sub>n</sub><sup>QST</sup> ∩ A<sub>n</sub><sup>NE</sup> ∩ A<sub>n</sub><sup>ABS</sup>|</th></tr></thead><tbody><tr><td>1</td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>2</td><td>12 = 1.2 · 10<sup>1</sup></td><td>6 = 6 · 10<sup>0</sup></td><td>8 = 8 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td></tr><tr><td>3</td><td>34 = 3.4 · 10<sup>1</sup></td><td>16 = 1.6 · 10<sup>1</sup></td><td>24 = 2.4 · 10<sup>1</sup></td><td>12 = 1.2 · 10<sup>1</sup></td></tr><tr><td>4</td><td>94 = 9.4 · 10<sup>1</sup></td><td>44 = 4.4 · 10<sup>1</sup></td><td>68 = 6.8 · 10<sup>1</sup></td><td>32 = 3.2 · 10<sup>1</sup></td></tr><tr><td>5</td><td>258 = 2.58 · 10<sup>2</sup></td><td>120 = 1.2 · 10<sup>2</sup></td><td>188 = 1.88 · 10<sup>2</sup></td><td>88 = 8.8 · 10<sup>1</sup></td></tr><tr><td>6</td><td>706 = 7.06 · 10<sup>2</sup></td><td>328 = 3.28 · 10<sup>2</sup></td><td>516 = 5.16 · 10<sup>2</sup></td><td>240 = 2.4 · 10<sup>2</sup></td></tr><tr><td>7</td><td>1,930 = 1.93 · 10<sup>3</sup></td><td>896 = 8.96 · 10<sup>2</sup></td><td>1,412 = 1.412 · 10<sup>3</sup></td><td>656 = 6.56 · 10<sup>2</sup></td></tr><tr><td>8</td><td>5,274 = 5.274 · 10<sup>3</sup></td><td>2,448 = 2.448 · 10<sup>3</sup></td><td>3,860 = 3.86 · 10<sup>3</sup></td><td>1,792 = 1.792 · 10<sup>3</sup></td></tr><tr><td>9</td><td>14,410 = 1.441 · 10<sup>4</sup></td><td>6,688 = 6.688 · 10<sup>3</sup></td><td>10,548 = 1.0548 · 10<sup>4</sup></td><td>4,896 = 4.896 · 10<sup>3</sup></td></tr><tr><td>10</td><td>39,370 = 3.937 · 10<sup>4</sup></td><td>18,272 = 1.8272 · 10<sup>4</sup></td><td>28,820 = 2.882 · 10<sup>4</sup></td><td>13,376 = 1.3376 · 10<sup>4</sup></td></tr><tr><td>11</td><td>≈ 1.0756 · 10<sup>5</sup></td><td>49,920 = 4.992 · 10<sup>4</sup></td><td>78,740 = 7.874 · 10<sup>4</sup></td><td>36,544 = 3.6544 · 10<sup>4</sup></td></tr><tr><td>12</td><td>≈ 2.9387 · 10<sup>5</sup></td><td>≈ 1.3638 · 10<sup>5</sup></td><td>≈ 2.1512 · 10<sup>5</sup></td><td>99,840 = 9.984 · 10<sup>4</sup></td></tr><tr><td>13</td><td>≈ 8.0286 · 10<sup>5</sup></td><td>≈ 3.7261 · 10<sup>5</sup></td><td>≈ 5.8773 · 10<sup>5</sup></td><td>≈ 2.7277 · 10<sup>5</sup></td></tr></tbody></table>

Enumeration of some classes of discrete aggregation functions for $n \le 13$. From top to bottom, and left to right: discrete aggregation functions in general, given by Proposition 3.1; commutative discrete aggregation functions, given by Proposition 3.2; bisymmetric and quasitrivial discrete aggregation functions, bisymmetric and quasitrivial discrete aggregation functions with neutral element and bisymmetric and quasitrivial discrete aggregation functions with absorbing element, given by Proposition 3.4; and finally associative and quasitrivial discrete aggregation functions, associative and quasitrivial discrete aggregation functions with neutral element, associative and quasitrivial discrete aggregation functions with absorbing element, and associative and quasitrivial discrete aggregation functions with neutral and absorbing elements, given by Proposition 3.3.

(iii) ([28, Proposition 4.6]) The number of bisymmetric and quasitrivial discrete aggregation functions with absorbing element, $|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}|$, satisfies the recurrence equation

$$|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}| = |A_n^{BSM} \cap A_n^{QST}| - 2, \quad (20)$$

for all $n \ge 1$, with $|A_1^{BSM} \cap A_1^{QST} \cap A_1^{ABS}| = 2$. Therefore, its general formula is given by

$$|A_n^{BSM} \cap A_n^{QST} \cap A_n^{ABS}| = 3 \cdot 2^n - 4. \quad (21)$$

In Table 2 we list the first 13 values of all the formulas that have appeared in this section.

## 3.2. Discrete conjunctions and disjunctions

As noteworthy examples of discrete aggregation functions, we have the so-called discrete conjunctions and discrete disjunctions.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Definition 3.5.** ([22]) A discrete conjunction is a discrete aggregation function $C: L_n^2 \to L_n$ with boundary conditions $C(0, n) = C(n, 0) = 0$.

From this definition, it is straightforward to verify that every discrete conjunction has 0 as absorbing element. On the other hand, we have the concept of discrete disjunction, which has $n$ as absorbing element.

**Definition 3.6.** ([22]) A discrete disjunction is a discrete aggregation function $D: L_n^2 \to L_n$ with boundary conditions $D(0, n) = D(n, 0) = n$.

**Remark 3.3.** Some sources in the literature, such as [27, 40, 41], consider conjunctions as those aggregation functions that are upper bounded by the minimum. However, in this paper (as in [22]) we have chosen to give a more generic definition, which encompasses the definition given in [27, 40, 41]. In the same way, in these sources disjunctions are considered as those aggregation functions which are lower bounded by the maximum.

The discovery of a general formula for the number of discrete conjunctions is attributed to Munar et al. in [19]. Similar to the formulas involved in Propositions 3.1 and 3.2 for discrete aggregation functions, they again used the concept of plane partition to derive the formula for their enumeration. For conjunctions, it is the following one.

**Proposition 3.5.** ([19, Proposition 4.3]) Let $n \ge 1$, and let us denote by $C_n$ the set of discrete conjunctions defined on $L_n$. Then,

$$|C_n| = \prod_{i=1}^{n} \prod_{j=1}^{n} \prod_{k=1}^{n} \frac{i+j+k-1}{i+j+k-2} - \prod_{i=1}^{n} \prod_{j=1}^{n} \prod_{k=1}^{n-1} \frac{i+j+k-1}{i+j+k-2} \quad (22)$$

They derived this formula by employing the same procedure used to obtain the formula of Proposition 3.1. The set $C_n$ is decomposed into mutually disjoint sets, each of which corresponds to a specific set of $(a, b, c)$-cubic plane partitions. Similarly, the formula for the enumeration of commutative discrete conjunctions is obtained by a similar process, but based on Equation (7), which provides the number of symmetric $(a, a, c)$-cubic plane partitions.

**Proposition 3.6.** ([19, Proposition 4.5]) Let $n \ge 1$, and let us denote by $C_n^{\text{COM}}$ the set of commutative discrete conjunctions defined on $L_n$. Then,

$$|C_n^{\text{COM}}| = \prod_{i=1}^{n} \left( \frac{2i+n-1}{2i-1} \cdot \prod_{j=i+1}^{n} \frac{i+j+n-1}{i+j-1} \right) - \prod_{i=1}^{n} \left( \frac{2i+n-2}{2i-1} \cdot \prod_{j=i+1}^{n} \frac{i+j+n-2}{i+j-1} \right) \quad (23)$$

Still building upon the framework of plane partitions, Bejines and Ojeda-Hernández presented in [42] a formula for the number of discrete conjunctions having neutral element $n$, which they termed *semicopulas* from the perspective of *totally symmetric plane partitions*, originally introduced by Mills et al. in [43]. It is worth highlighting that the literature encompasses a significant variety of plane partition types and related structures (refer to [44] for more information). The authors devoted a whole paper to establishing a bijection between discrete conjunctions with neutral element $n$ and these plane partitions, so we omit the details of this bijection. The formula under discussion is as follows.

**Proposition 3.7.** ([42, Corollary 17]) Let $n \ge 1$, and let us denote by $C_n^{\text{NE}(n)}$ the set of discrete conjunctions with neutral element $n$ defined on $L_n$. Then,

$$|C_n^{\text{NE}(n)}| = \prod_{i=0}^{n-1} \frac{(3i+1)(6i)! (2i)!}{(4i+1)!(4i)!} \quad (24)$$

Another noteworthy property is smoothness, which was fully characterized for discrete conjunctions operators by Munar et al. in [20] using the concept of alternating sign matrix. This type of matrices were introduced by Robbins and Rumsey in [45].

**Definition 3.7.** ([45, Definition 3.1]) An alternating sign matrix is an $n \times n$ matrix $A = (a_{i,j})$, which satisfies the following properties:
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Definition 3.5.** ([22]) A discrete conjunction is a discrete aggregation function $C: L_n^2 \to L_n$ with boundary conditions $C(0, n) = C(n, 0) = 0$.

From this definition, it is straightforward to verify that every discrete conjunction has 0 as absorbing element. On the other hand, we have the concept of discrete disjunction, which has $n$ as absorbing element.

**Definition 3.6.** ([22]) A discrete disjunction is a discrete aggregation function $D: L_n^2 \to L_n$ with boundary conditions $D(0, n) = D(n, 0) = n$.

**Remark 3.3.** Some sources in the literature, such as [27, 40, 41], consider conjunctions as those aggregation functions that are upper bounded by the minimum. However, in this paper (as in [22]) we have chosen to give a more generic definition, which encompasses the definition given in [27, 40, 41]. In the same way, in these sources disjunctions are considered as those aggregation functions which are lower bounded by the maximum.

The discovery of a general formula for the number of discrete conjunctions is attributed to Munar et al. in [19]. Similar to the formulas involved in Propositions 3.1 and 3.2 for discrete aggregation functions, they again used the concept of plane partition to derive the formula for their enumeration. For conjunctions, it is the following one.

**Proposition 3.5.** ([19, Proposition 4.3]) Let $n \ge 1$, and let us denote by $C_n$ the set of discrete conjunctions defined on $L_n$. Then,

$$|C_n| = \prod_{i=1}^{n} \prod_{j=1}^{n} \prod_{k=1}^{n} \frac{i+j+k-1}{i+j+k-2} - \prod_{i=1}^{n} \prod_{j=1}^{n} \prod_{k=1}^{n-1} \frac{i+j+k-1}{i+j+k-2} \quad (22)$$

They derived this formula by employing the same procedure used to obtain the formula of Proposition 3.1. The set $C_n$ is decomposed into mutually disjoint sets, each of which corresponds to a specific set of $(a, b, c)$-cubic plane partitions. Similarly, the formula for the enumeration of commutative discrete conjunctions is obtained by a similar process, but based on Equation (7), which provides the number of symmetric $(a, a, c)$-cubic plane partitions.

**Proposition 3.6.** ([19, Proposition 4.5]) Let $n \ge 1$, and let us denote by $C_n^{COM}$ the set of commutative discrete conjunctions defined on $L_n$. Then,

$$|C_n^{COM}| = \prod_{i=1}^{n} \left( \frac{2i + n - 1}{2i - 1} \cdot \prod_{j=i+1}^{n} \frac{i + j + n - 1}{i + j - 1} \right) - \prod_{i=1}^{n} \left( \frac{2i + n - 2}{2i - 1} \cdot \prod_{j=i+1}^{n} \frac{i + j + n - 2}{i + j - 1} \right) \quad (23)$$

Still building upon the framework of plane partitions, Bejines and Ojeda-Hernández presented in [42] a formula for the number of discrete conjunctions having neutral element $n$, which they termed *semicopulas* from the perspective of *totally symmetric plane partitions*, originally introduced by Mills et al. in [43]. It is worth highlighting that the literature encompasses a significant variety of plane partition types and related structures (refer to [44] for more information). The authors devoted a whole paper to establishing a bijection between discrete conjunctions with neutral element $n$ and these plane partitions, so we omit the details of this bijection. The formula under discussion is as follows.

**Proposition 3.7.** ([42, Corollary 17]) Let $n \ge 1$, and let us denote by $C_n^{NE(n)}$ the set of discrete conjunctions with neutral element $n$ defined on $L_n$. Then,

$$|C_n^{NE(n)}| = \prod_{i=0}^{n-1} \frac{(3i + 1)(6i)! (2i)!}{(4i + 1)! (4i)!} \quad (24)$$

Another noteworthy property is smoothness, which was fully characterized for discrete conjunctions operators by Munar et al. in [20] using the concept of alternating sign matrix. This type of matrices were introduced by Robbins and Rumsey in [45].

**Definition 3.7.** ([45, Definition 3.1]) An alternating sign matrix is an $n \times n$ matrix $A = (a_{i,j})$, which satisfies the following properties:
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

$$ (\text{ASM1}) \quad a_{i,j} \in \{-1, 0, 1\}, \text{ for all } i, j \in \{1, \dots, n\}. $$

$$ (\text{ASM2}) \quad \sum_{i=1}^{n} a_{i,k} = \sum_{j=1}^{n} a_{l,j} = 1, \text{ for all } k, l \in \{1, \dots, n\}. $$

(ASM3) In every row and column the non-zero entries alternate in sign.

**Theorem 3.1.** ([20, Theorem 3.2]) Let $n \ge 1$, and let $C: L_n^2 \to L_n$ be a binary operator. Then, $C$ is a smooth discrete conjunction if, and only if, there exists an $n \times n$ alternating sign matrix $A = (a_{i,j})$ such that

$$ C(x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ \sum_{i=1}^{x} \sum_{j=1}^{y} a_{i,j}, & \text{otherwise,} \end{cases} \tag{25} $$

for all $x, y \in L_n$.

The characterization in question establishes a bijection between $C_n^{\text{SMT}}$, the set of smooth discrete conjunctions on $L_n$, and $M_n$, the set of $n \times n$ alternating sign matrices. Consequently, possessing knowledge of the cardinality of $M_n$ would suffice for determining the cardinality of $C_n^{\text{SMT}}$. Robbins and Rumsey conjectured a formula for the number of alternating sign matrices. Later, in [46], Zeilberger proved the conjecture through intricate arguments, a fact underscored by the substantial length of the paper, spanning 84 pages, and the meticulous review conducted by 88 reviewers. In between, there were other attempts to prove this conjecture, we refer to [47] for an explanation of the evolution of the problem up to its proof. This formula is given in Proposition 3.9.

Additionally, there is a deeper dimension to this formula. In [46], Zeilberger extended his study by establishing a correspondence between the enumeration of alternating sign matrices and that of n-Gogs and n-Magogs, which are triangular configurations of integers. Subsequently, Munar et al. in [20] introduced another category of triangular groupings, which they termed rectified n-Magogs, and can be defined as the set:

$$ \mathcal{R}\mathcal{T}_n = \{ \hat{t} = (\hat{t}_{i,j}), 1 \le i \le n, 1 \le j \le n - i + 1 \mid 0 \le \hat{t}_{i,j} \le j, \hat{t}_{i,j} \ge \hat{t}_{i+1,j}, \hat{t}_{i,j} \le \hat{t}_{i,j+1} \}. \tag{26} $$

They proved two important properties of rectified n-Magogs: their cardinality, and a bijection with the set of commutative discrete conjunctions with neutral element $n$.

**Proposition 3.8.** Let $n \ge 1$, and let us denote by

* $\mathcal{R}\mathcal{T}_n$, the set of rectified n-Magogs;

* $M_n$, the set of $n \times n$ alternating sign matrices;

* $C_n^{\text{COM}}$, the set of commutative discrete conjunctions defined on $L_n$;

* $C_n^{\text{NE}(n)}$, the set of discrete conjunctions with neutral element $n$ defined on $L_n$.

Then,

$$ (i) \quad ([20, \text{Lemma 4.1}]) \quad |\mathcal{R}\mathcal{T}_n| = |\mathcal{M}_{n+1}|. $$

$$ (ii) \quad ([20, \text{Lemma 4.2}]) \quad \text{The transformation } \omega: \mathcal{R}\mathcal{T}_n \to C_{n+1}^{\text{COM}} \cap C_{n+1}^{\text{NE}(n)}, \text{ given by} $$

$$ \omega(\hat{t}) (x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ x, & \text{if } y = n + 1, \\ y, & \text{if } x = n + 1, \\ \hat{t}_{n-y,x}, & \text{if } 1 \le x \le n \text{ and } x \le y \le n, \\ \hat{t}_{n-x,y}, & \text{if } 2 \le x \le n \text{ and } 1 \le y < x, \end{cases} \tag{27} $$

for all $x, y \in L_{n+1}$, is a bijection.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

$$ (\text{ASM1}) \quad a_{i,j} \in \{-1, 0, 1\}, \text{ for all } i, j \in \{1, \dots, n\}. $$

$$ (\text{ASM2}) \quad \sum_{i=1}^{n} a_{i,k} = \sum_{j=1}^{n} a_{l,j} = 1, \text{ for all } k, l \in \{1, \dots, n\}. $$

(ASM3) In every row and column the non-zero entries alternate in sign.

**Theorem 3.1.** ([20, Theorem 3.2]) Let $n \ge 1$, and let $C: L_n^2 \to L_n$ be a binary operator. Then, $C$ is a smooth discrete conjunction if, and only if, there exists an $n \times n$ alternating sign matrix $A = (a_{i,j})$ such that

$$ C(x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ \sum_{i=1}^{x} \sum_{j=1}^{y} a_{i,j}, & \text{otherwise,} \end{cases} \qquad (25) $$

for all $x, y \in L_n$.

The characterization in question establishes a bijection between $C_n^{\text{SMT}}$, the set of smooth discrete conjunctions on $L_n$, and $M_n$, the set of $n \times n$ alternating sign matrices. Consequently, possessing knowledge of the cardinality of $M_n$ would suffice for determining the cardinality of $C_n^{\text{SMT}}$. Robbins and Rumsey conjectured a formula for the number of alternating sign matrices. Later, in [46], Zeilberger proved the conjecture through intricate arguments, a fact underscored by the substantial length of the paper, spanning 84 pages, and the meticulous review conducted by 88 reviewers. In between, there were other attempts to prove this conjecture, we refer to [47] for an explanation of the evolution of the problem up to its proof. This formula is given in Proposition 3.9.

Additionally, there is a deeper dimension to this formula. In [46], Zeilberger extended his study by establishing a correspondence between the enumeration of alternating sign matrices and that of n-Gogs and n-Magogs, which are triangular configurations of integers. Subsequently, Munar et al. in [20] introduced another category of triangular groupings, which they termed rectified n-Magogs, and can be defined as the set:

$$ \mathcal{R}\mathcal{T}_n = \{ \hat{t} = (\hat{t}_{i,j}), 1 \le i \le n, 1 \le j \le n - i + 1 \mid 0 \le \hat{t}_{i,j} \le j, \hat{t}_{i,j} \ge \hat{t}_{i+1,j}, \hat{t}_{i,j} \le \hat{t}_{i,j+1} \}. \qquad (26) $$

They proved two important properties of rectified n-Magogs: their cardinality, and a bijection with the set of commutative discrete conjunctions with neutral element $n$.

**Proposition 3.8.** Let $n \ge 1$, and let us denote by

* $\mathcal{R}\mathcal{T}_n$, the set of rectified n-Magogs;

* $M_n$, the set of $n \times n$ alternating sign matrices;

* $C_n^{\text{COM}}$, the set of commutative discrete conjunctions defined on $L_n$;

* $C_n^{\text{NE}(n)}$, the set of discrete conjunctions with neutral element $n$ defined on $L_n$.

Then,

$$ (i) \quad ([20, \text{Lemma 4.1}]) |\mathcal{R}\mathcal{T}_n| = |M_{n+1}|. $$

$$ (ii) \quad ([20, \text{Lemma 4.2}]) \quad \text{The transformation } \omega: \mathcal{R}\mathcal{T}_n \to C_{n+1}^{\text{COM}} \cap C_{n+1}^{\text{NE}(n)}, \text{ given by} $$

$$ \omega(\hat{t}) (x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ x, & \text{if } y = n + 1, \\ y, & \text{if } x = n + 1, \\ \hat{t}_{n-y,x}, & \text{if } 1 \le x \le n \text{ and } x \le y \le n, \\ \hat{t}_{n-x,y}, & \text{if } 2 \le x \le n \text{ and } 1 \le y < x, \end{cases} \qquad (27) $$

for all $x, y \in L_{n+1}$, is a bijection.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

The establishment of $|C_{n+1}^{COM} \cap C_{n+1}^{NE(n)}|$ is straightforward from Proposition 3.8. It is important to highlight, however, that despite $C_n^{COM} \cap C_n^{NE(n)}$ and $M_n$ share the same enumeration, a bijection between these two sets remains unknown. This is partly attributed to the absence of a known bijection between the sets of n-Gogs and n-Magogs as well.

**Proposition 3.9.** ([20, 46]) Let $n \ge 2$, and let us denote by

* $M_n$, the set of $n \times n$ alternating sign matrices;

* $C_n^{SMT}$, the set of smooth discrete conjunctions defined on $L_n$;

* $C_n^{COM}$, the set of commutative discrete conjunctions defined on $L_n$;

* $C_n^{NE(n)}$, the set of discrete conjunctions with neutral element $n$ defined on $L_n$.

Then,

$$|C_n^{COM} \cap C_n^{NE(n)}| = |C_n^{SMT}| = |M_n| = \prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!} = \prod_{i=1}^{n} \prod_{j=i}^{n} \frac{n+i+j-1}{2i+j-1} \quad (28)$$

**Remark 3.4.** Although $|C_n^{COM} \cap C_n^{NE(n)}| = |C_n^{SMT}|$, it should be noted that in general $C_n^{COM} \cap C_n^{NE(n)} \ne C_n^{SMT}$, as indicated in [20, Observation 3].

To conclude our discussion on discrete conjunctions, we revisit the result that provides an enumeration for smooth and associative discrete conjunctions. In certain instances, the smoothness requirement serves to alleviate the complexity inherent to associativity.

**Proposition 3.10.** ([20, Corollary 3.7.2]) Let $n \ge 1$ and let us denote by $C_n^{SMT}$ and $C_n^{ASV}$ the sets of smooth discrete conjunctions and associative discrete conjunctions, respectively, defined on $L_n$. Then,

$$|C_n^{SMT} \cap C_n^{ASV}| = 2^{n-1}. \quad (29)$$

Let us now consider the class of discrete disjunctions. An approach for determining their enumeration is to again consider plane partitions. However, it is worth noting that in [19, Proposition 4.3] it was established that discrete conjunctions can be bijectively transformed into discrete disjunctions. Therefore, the formulas mentioned above for discrete conjunctions and some of their subclasses can be applied to enumerate discrete disjunctions and some of their subclasses. This bijection is constructed based on the unique strong discrete negation, $N_C$, which is an automorphism on $L_n$.

**Proposition 3.11.** ([19, Proposition 4.3]) Let $n \ge 1$, and let us denote by $C_n$ and $D_n$ the sets of discrete conjunctions and discrete disjunctions, respectively, defined on $L_n$. Let $\varphi: C_n \to D_n$ be defined point-wise by

$$\varphi(C)(x, y) = n - C(n - x, n - y),$$

for all $x, y \in L_n$, where $C \in C_n$. Then, $\varphi(C)$ is a discrete disjunction, and the transformation $\varphi$ is a bijection.

This bijection makes it possible to seamlessly use the cardinality of one class for the other, as follows:

* Discrete disjunctions, $D_n$. Its cardinality coincides with $|C_n|$, given in Proposition 3.5.

* Commutative discrete disjunctions, $D_n^{COM}$. Its cardinality coincides with $|C_n^{COM}|$, given in Proposition 3.6.

* Discrete disjunctions with neutral element 0, $D_n^{NE(0)}$. Its cardinality coincides with the cardinality of discrete conjunctions with neutral element $n$, $C_n^{NE(n)}$, given in Proposition 3.7.

* Smooth discrete disjunctions, $D_n^{SMT}$. Its cardinality coincides with $|C_n^{SMT}|$, given in Proposition 3.9.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

The establishment of $|C_{n+1}^{COM} \cap C_{n+1}^{NE(n)}|$ is straightforward from Proposition 3.8. It is important to highlight, however, that despite $C_n^{COM} \cap C_n^{NE(n)}$ and $M_n$ share the same enumeration, a bijection between these two sets remains unknown. This is partly attributed to the absence of a known bijection between the sets of n-Gogs and n-Magogs as well.

**Proposition 3.9.** ([20, 46]) Let $n \ge 2$, and let us denote by

* $M_n$, the set of $n \times n$ alternating sign matrices;

* $C_n^{SMT}$, the set of smooth discrete conjunctions defined on $L_n$;

* $C_n^{COM}$, the set of commutative discrete conjunctions defined on $L_n$;

* $C_n^{NE(n)}$, the set of discrete conjunctions with neutral element $n$ defined on $L_n$.

Then,

$$|C_n^{COM} \cap C_n^{NE(n)}| = |C_n^{SMT}| = |M_n| = \prod_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!} = \prod_{i=1}^{n} \prod_{j=i}^{n} \frac{n+i+j-1}{2i+j-1} \quad (28)$$

**Remark 3.4.** Although $|C_n^{COM} \cap C_n^{NE(n)}| = |C_n^{SMT}|$, it should be noted that in general $C_n^{COM} \cap C_n^{NE(n)} \ne C_n^{SMT}$, as indicated in [20, Observation 3].

To conclude our discussion on discrete conjunctions, we revisit the result that provides an enumeration for smooth and associative discrete conjunctions. In certain instances, the smoothness requirement serves to alleviate the complexity inherent to associativity.

**Proposition 3.10.** ([20, Corollary 3.7.2]) Let $n \ge 1$ and let us denote by $C_n^{SMT}$ and $C_n^{ASV}$ the sets of smooth discrete conjunctions and associative discrete conjunctions, respectively, defined on $L_n$. Then,

$$|C_n^{SMT} \cap C_n^{ASV}| = 2^{n-1}. \quad (29)$$

Let us now consider the class of discrete disjunctions. An approach for determining their enumeration is to again consider plane partitions. However, it is worth noting that in [19, Proposition 4.3] it was established that discrete conjunctions can be bijectively transformed into discrete disjunctions. Therefore, the formulas mentioned above for for discrete conjunctions and some of their subclasses can be applied to enumerate discrete disjunctions and some of their subclasses. This bijection is constructed based on the unique strong discrete negation, $N_C$, which is an automorphism on $L_n$.

**Proposition 3.11.** ([19, Proposition 4.3]) Let $n \ge 1$, and let us denote by $C_n$ and $D_n$ the sets of discrete conjunctions and discrete disjunctions, respectively, defined on $L_n$. Let $\varphi: C_n \to D_n$ be defined point-wise by

$$\varphi(C)(x, y) = n - C(n - x, n - y),$$

for all $x, y \in L_n$, where $C \in C_n$. Then, $\varphi(C)$ is a discrete disjunction, and the transformation $\varphi$ is a bijection.

This bijection makes it possible to seamlessly use the cardinality of one class for the other, as follows:

* Discrete disjunctions, $D_n$. Its cardinality coincides with $|C_n|$, given in Proposition 3.5.

* Commutative discrete disjunctions, $D_n^{COM}$. Its cardinality coincides with $|C_n^{COM}|$, given in Proposition 3.6.

* Discrete disjunctions with neutral element 0, $D_n^{NE(0)}$. Its cardinality coincides with the cardinality of discrete conjunctions with neutral element $n$, $C_n^{NE(n)}$, given in Proposition 3.7.

* Smooth discrete disjunctions, $D_n^{SMT}$. Its cardinality coincides with $|C_n^{SMT}|$, given in Proposition 3.9.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>Table 3</caption><thead><tr><th>n</th><th>|C<sub>n</sub>|, |D<sub>n</sub>|</th><th>|C<sup>COM</sup><sub>n</sub>|, |D<sup>COM</sup><sub>n</sub>|</th><th>C<sup>NE(n)</sup><sub>n</sub>, D<sup>NE(0)</sup><sub>n</sub></th><th>|C<sup>SMT</sup><sub>n</sub>|, |D<sup>SMT</sup><sub>n</sub>|</th><th>|C<sup>SMT</sup><sub>n</sub> ∩ C<sup>ASV</sup><sub>n</sub>|, |D<sup>SMT</sup><sub>n</sub> ∩ D<sup>ASV</sup><sub>n</sub>|</th></tr></thead><tbody><tr><td>1</td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td></tr><tr><td>2</td><td>14 = 1.4 · 10<sup>1</sup></td><td>6 = 6 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>3</td><td>805 = 8.05 · 10<sup>2</sup></td><td>77 = 7.7 · 10<sup>1</sup></td><td>11 = 1.1 · 10<sup>1</sup></td><td>7 = 7 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td></tr><tr><td>4</td><td>≈ 2.0815 · 10<sup>5</sup></td><td>2,100 = 2.1 · 10<sup>3</sup></td><td>170 = 1.7 · 10<sup>2</sup></td><td>42 = 4.2 · 10<sup>1</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>5</td><td>≈ 2.5041 · 10<sup>8</sup></td><td>≈ 1.2269 · 10<sup>5</sup></td><td>7,429 = 7.429 · 10<sup>3</sup></td><td>429 = 4.29 · 10<sup>2</sup></td><td>16 = 1.6 · 10<sup>1</sup></td></tr><tr><td>6</td><td>≈ 1.4234 · 10<sup>12</sup></td><td>≈ 1.5459 · 10<sup>7</sup></td><td>≈ 9.2046 · 10<sup>5</sup></td><td>7,436 = 7.436 · 10<sup>3</sup></td><td>32 = 3.2 · 10<sup>1</sup></td></tr><tr><td>7</td><td>≈ 3.8534 · 10<sup>16</sup></td><td>≈ 4.2204 · 10<sup>9</sup></td><td>≈ 3.238 · 10<sup>8</sup></td><td>≈ 2.1835 · 10<sup>5</sup></td><td>64 = 6.4 · 10<sup>1</sup></td></tr><tr><td>8</td><td>≈ 4.9888 · 10<sup>21</sup></td><td>≈ 2.5045 · 10<sup>12</sup></td><td>≈ 3.2367 · 10<sup>11</sup></td><td>≈ 1.085 · 10<sup>7</sup></td><td>128 = 1.28 · 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 3.0961 · 10<sup>27</sup></td><td>≈ 3.238 · 10<sup>15</sup></td><td>≈ 9.1986 · 10<sup>14</sup></td><td>≈ 9.1184 · 10<sup>8</sup></td><td>256 = 2.56 · 10<sup>2</sup></td></tr><tr><td>10</td><td>≈ 9.2223 · 10<sup>33</sup></td><td>≈ 9.1352 · 10<sup>18</sup></td><td>≈ 7.4347 · 10<sup>18</sup></td><td>≈ 1.2953 · 10<sup>11</sup></td><td>512 = 5.12 · 10<sup>2</sup></td></tr><tr><td>11</td><td>≈ 1.3195 · 10<sup>41</sup></td><td>≈ 5.6307 · 10<sup>22</sup></td><td>≈ 1.7094 · 10<sup>23</sup></td><td>≈ 3.1096 · 10<sup>13</sup></td><td>1,024 = 1.024 · 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 9.0708 · 10<sup>48</sup></td><td>≈ 7.5891 · 10<sup>26</sup></td><td>≈ 1.1183 · 10<sup>28</sup></td><td>≈ 1.2611 · 10<sup>16</sup></td><td>2,048 = 2.048 · 10<sup>3</sup></td></tr><tr><td>13</td><td>≈ 2.997 · 10<sup>57</sup></td><td>≈ 2.2381 · 10<sup>31</sup></td><td>≈ 2.0819 · 10<sup>33</sup></td><td>≈ 8.6394 · 10<sup>18</sup></td><td>4,096 = 4.096 · 10<sup>3</sup></td></tr></tbody></table>

Enumeration of some classes of discrete conjunctions and disjunctions, for $n \le 13$. From left to right: discrete conjunctions and disjunctions in general, given by Proposition 3.5; commutative discrete conjunctions and disjunctions, given by Proposition 3.6; discrete conjunctions having neutral element $n$ and discrete disjunctions having neutral element 0, given by Proposition 3.7; smooth discrete conjunctions and disjunctions, given by Proposition 3.9; and finally smooth discrete conjunctions and disjunctions which are associative, given by Proposition 3.10.

* Commutative discrete disjunctions with neutral element 0, $D_n^{COM} \cap D_n^{NE(0)}$. Its cardinality coincides with the cardinality of commutative discrete conjunctions with neutral element $n$, $C_n^{COM} \cap C_n^{NE(n)}$, given in Proposition 3.9.

* Smooth and associative discrete disjunctions, $D_n^{SMT} \cap D_n^{ASV}$. Its cardinality coincides with $|C_n^{SMT} \cap C_n^{ASV}|$, given in Proposition 3.10.

In Table 3 we indicate the first 13 terms of the different enumerations for discrete conjunctions and disjunctions.

### 3.3. Discrete quasi-copulas and copulas

Discrete quasi-copulas and copulas are particular instances of discrete conjunctions. Quasi-copulas on [0, 1] were studied in the framework of theoretical statistics in [48], and on discrete settings, the concept of discrete quasi-copula was introduced in [49], and further studied in [21, 50].

**Definition 3.8.** ([21, 49, 50]) A discrete quasi-copula is a discrete conjunction $Q: L_n^2 \to L_n$ which is 1-Lipschitz and has $n$ as neutral element.

A comprehensive characterization of discrete quasi-copulas was originally presented by Aguiló et al. in [21], also from the point of view of alternating sign matrices. Intriguingly, they arrived at the same result as Theorem 3.1, yielding to the same enumeration as the one outlined in Proposition 3.9. It is worth noting that the work of Aguiló et al. in [21] chronologically predates the independent research conducted by Munar et al. in [20], and both aimed to characterize different operators whose definitions are also different. A noteworthy conclusion highlighted in [20] is that discrete quasi-copulas do not constitute a proper subset of the set of smooth discrete conjunctions (since 1-Lipschitz property implies smoothness), but rather that the classes of quasi-copulas and smooth discrete conjunctions coincide.

As a subset of discrete quasi-copulas, there is the concept of discrete copula, a notion that also finds extensive application in theoretical statistics. This significance is primarily attributable to Sklar's theorem, originally proposed on [0, 1] in [51], and its equivalent on $L_n$, stated in [52].

**Definition 3.9.** ([18]) A discrete copula is a discrete quasi-copula $K: L_n^2 \to L_n$ which is 2-increasing, i.e., $K(x_1, y_1) + K(x_2, y_2) \ge K(x_1, y_2) + K(x_2, y_1)$, for all $x_1, x_2, y_1, y_2 \in L_n$ whenever $x_1 \le x_2$ and $y_1 \le y_2$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 3**</caption><thead><tr><th>n</th><th>|C<sub>n</sub>|, |D<sub>n</sub>|</th><th>|C<sup>COM</sup><sub>n</sub>|, |D<sup>COM</sup><sub>n</sub>|</th><th>C<sup>NE(n)</sup><sub>n</sub>, |D<sup>NE(0)</sup><sub>n</sub>|</th><th>|C<sup>SMT</sup><sub>n</sub>|, |D<sup>SMT</sup><sub>n</sub>|</th><th>|C<sup>SMT</sup><sub>n</sub> &cap; C<sup>ASV</sup><sub>n</sub>|, |D<sup>SMT</sup><sub>n</sub> &cap; D<sup>ASV</sup><sub>n</sub>|</th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>14 = 1.4 &middot; 10<sup>1</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>805 = 8.05 &middot; 10<sup>2</sup></td><td>77 = 7.7 &middot; 10<sup>1</sup></td><td>11 = 1.1 &middot; 10<sup>1</sup></td><td>7 = 7 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td></tr><tr><td>4</td><td>≈ 2.0815 &middot; 10<sup>5</sup></td><td>2,100 = 2.1 &middot; 10<sup>3</sup></td><td>170 = 1.7 &middot; 10<sup>2</sup></td><td>42 = 4.2 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>0</sup></td></tr><tr><td>5</td><td>≈ 2.5041 &middot; 10<sup>8</sup></td><td>≈ 1.2269 &middot; 10<sup>5</sup></td><td>7,429 = 7.429 &middot; 10<sup>3</sup></td><td>429 = 4.29 &middot; 10<sup>2</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td></tr><tr><td>6</td><td>≈ 1.4234 &middot; 10<sup>12</sup></td><td>≈ 1.5459 &middot; 10<sup>7</sup></td><td>≈ 9.2046 &middot; 10<sup>5</sup></td><td>7,436 = 7.436 &middot; 10<sup>3</sup></td><td>32 = 3.2 &middot; 10<sup>1</sup></td></tr><tr><td>7</td><td>≈ 3.8534 &middot; 10<sup>16</sup></td><td>≈ 4.2204 &middot; 10<sup>9</sup></td><td>≈ 3.238 &middot; 10<sup>8</sup></td><td>≈ 2.1835 &middot; 10<sup>5</sup></td><td>64 = 6.4 &middot; 10<sup>1</sup></td></tr><tr><td>8</td><td>≈ 4.9888 &middot; 10<sup>21</sup></td><td>≈ 2.5045 &middot; 10<sup>12</sup></td><td>≈ 3.2367 &middot; 10<sup>11</sup></td><td>≈ 1.085 &middot; 10<sup>7</sup></td><td>128 = 1.28 &middot; 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 3.0961 &middot; 10<sup>27</sup></td><td>≈ 3.238 &middot; 10<sup>15</sup></td><td>≈ 9.1986 &middot; 10<sup>14</sup></td><td>≈ 9.1184 &middot; 10<sup>8</sup></td><td>256 = 2.56 &middot; 10<sup>2</sup></td></tr><tr><td>10</td><td>≈ 9.2223 &middot; 10<sup>33</sup></td><td>≈ 9.1352 &middot; 10<sup>18</sup></td><td>≈ 7.4347 &middot; 10<sup>18</sup></td><td>≈ 1.2953 &middot; 10<sup>11</sup></td><td>512 = 5.12 &middot; 10<sup>2</sup></td></tr><tr><td>11</td><td>≈ 1.3195 &middot; 10<sup>41</sup></td><td>≈ 5.6307 &middot; 10<sup>22</sup></td><td>≈ 1.7094 &middot; 10<sup>23</sup></td><td>≈ 3.1096 &middot; 10<sup>13</sup></td><td>1,024 = 1.024 &middot; 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 9.0708 &middot; 10<sup>48</sup></td><td>≈ 7.5891 &middot; 10<sup>26</sup></td><td>≈ 1.1183 &middot; 10<sup>28</sup></td><td>≈ 1.2611 &middot; 10<sup>16</sup></td><td>2,048 = 2.048 &middot; 10<sup>3</sup></td></tr><tr><td>13</td><td>≈ 2.997 &middot; 10<sup>57</sup></td><td>≈ 2.2381 &middot; 10<sup>31</sup></td><td>≈ 2.0819 &middot; 10<sup>33</sup></td><td>≈ 8.6394 &middot; 10<sup>18</sup></td><td>4,096 = 4.096 &middot; 10<sup>3</sup></td></tr></tbody></table>

Enumeration of some classes of discrete conjunctions and disjunctions, for $n \le 13$. From left to right: discrete conjunctions and disjunctions in general, given by Proposition 3.5; commutative discrete conjunctions and disjunctions, given by Proposition 3.6; discrete conjunctions having neutral element $n$ and discrete disjunctions having neutral element 0, given by Proposition 3.7; smooth discrete conjunctions and disjunctions, given by Proposition 3.9; and finally smooth discrete conjunctions and disjunctions which are associative, given by Proposition 3.10.

* Commutative discrete disjunctions with neutral element 0, $D_n^{COM} \cap D_n^{NE(0)}$. Its cardinality coincides with the cardinality of commutative discrete conjunctions with neutral element $n$, $C_n^{COM} \cap C_n^{NE(n)}$, given in Proposition 3.9.

* Smooth and associative discrete disjunctions, $D_n^{SMT} \cap D_n^{ASV}$. Its cardinality coincides with $|C_n^{SMT} \cap C_n^{ASV}|$, given in Proposition 3.10.

In Table 3 we indicate the first 13 terms of the different enumerations for discrete conjunctions and disjunctions.

### 3.3. Discrete quasi-copulas and copulas

Discrete quasi-copulas and copulas are particular instances of discrete conjunctions. Quasi-copulas on [0, 1] were studied in the framework of theoretical statistics in [48], and on discrete settings, the concept of discrete quasi-copula was introduced in [49], and further studied in [21, 50].

**Definition 3.8.** ([21, 49, 50]) A discrete quasi-copula is a discrete conjunction $Q: L_n^2 \to L_n$ which is 1-Lipschitz and has $n$ as neutral element.

A comprehensive characterization of discrete quasi-copulas was originally presented by Aguiló et al. in [21], also from the point of view of alternating sign matrices. Intriguingly, they arrived at the same result as Theorem 3.1, yielding to the same enumeration as the one outlined in Proposition 3.9. It is worth noting that the work of Aguiló et al. in [21] chronologically predates the independent research conducted by Munar et al. in [20], and both aimed to characterize different operators whose definitions are also different. A noteworthy conclusion highlighted in [20] is that discrete quasi-copulas do not constitute a proper subset of the set of smooth discrete conjunctions (since 1-Lipschitz property implies smoothness), but rather that the classes of quasi-copulas and smooth discrete conjunctions coincide.

As a subset of discrete quasi-copulas, there is the concept of discrete copula, a notion that also finds extensive application in theoretical statistics. This significance is primarily attributable to Sklar's theorem, originally proposed on [0, 1] in [51], and its equivalent on $L_n$, stated in [52].

**Definition 3.9.** ([18]) A discrete copula is a discrete quasi-copula $K: L_n^2 \to L_n$ which is 2-increasing, i.e., $K(x_1, y_1) + K(x_2, y_2) \ge K(x_1, y_2) + K(x_2, y_1)$, for all $x_1, x_2, y_1, y_2 \in L_n$ whenever $x_1 \le x_2$ and $y_1 \le y_2$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Mayor et al. in [18] proposed a complete characterization of discrete copulas using permutation matrices, which are in fact a particular case of alternating sign matrices. The construction of a permutation matrix is derived from the permutation group of n elements, denoted as $\mathfrak{C}_n$.

**Definition 3.10.** ([53, Section 5.6]) A permutation matrix $P = (p_{i,j})$ is an $n \times n$ matrix such that there exists a permutation $\sigma \in \mathfrak{C}_n$ of $L_n \setminus \{0\}$ whereby $P$ is expressed as

$$p_{ij} = \begin{cases} 1, & \text{if } i = \sigma(j), \\ 0, & \text{otherwise,} \end{cases} \qquad (30)$$

for all $i, j \in \{1, \dots, n\}$.

They characterized discrete copulas using these matrices, yielding a result akin to Theorem 3.1. Subsequently, they were able to determine the cardinality of this class of discrete operators.

**Proposition 3.12.** ([18, Proposition 6]) Let $n \ge 1$, and let $K: L_n^2 \to L_n$ be a binary operator. Then, $K$ is a discrete copula if, and only if, there exists an $n \times n$ permutation matrix $P = (p_{ij})$ such that

$$K(x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ \sum_{i=1}^{x} \sum_{j=1}^{y} p_{ij}, & \text{otherwise,} \end{cases} \qquad (31)$$

for all $x, y \in L_n$.

**Proposition 3.13.** ([18, Corollary 1]) Let $n \ge 1$, and let us denote by $\mathcal{K}_n$ the set of discrete copulas defined on $L_n$. Then,

$$|\mathcal{K}_n| = n!. \qquad (32)$$

The same characterization of Proposition 3.12 also applies to commutative discrete copulas, but taking symmetric permutation matrices as the equivalent object.

**Proposition 3.14.** ([18, Corollary 1]) Let $n \ge 1$, and let us denote by $\mathcal{K}_n^{\text{COM}}$ the set of commutative discrete copulas defined on $L_n$. Then,

$$|\mathcal{K}_n^{\text{COM}}| = \sum_{i=0}^{\lfloor n/2 \rfloor} \frac{n!}{2^i (n-2i)! i!} \qquad (33)$$

In addition to these two results, Mayor et al. [18] also studied other properties of discrete copulas, such as the associative and idempotent-free properties.

**Proposition 3.15.** Let $n \ge 1$, and let us denote by $\mathcal{K}_n^{\text{ASV}}$ and $\mathcal{K}_n^{\text{IDF}}$ the sets of associative discrete copulas and idempotent-free discrete copulas, respectively, defined on $L_n$. Then, the following statements hold:

(i) ([18, Corollary 4]) $|\mathcal{K}_n^{\text{ASV}}| = 2^{n-1}$,

(ii) ([18, Proposition 8]) The only associative and idempotent-free discrete copula is the Łukasiewicz copula, given by $K_L(x, y) = \max\{0, x + y - n\}$, for all $x, y \in L_n$. Therefore, $|\mathcal{K}_n^{\text{ASV}} \cap \mathcal{K}_n^{\text{IDF}}| = 1$.

For discrete quasi-copulas and discrete copulas, we illustrate the first 13 terms of the different enumerations in Table 4. Although the enumeration of $Q_n$ coincides with that of $C_n^{\text{SMT}}$, since they are the same class of operators and has been illustrated previously in Table 3, we show it again so that it can be compared with the enumerations of discrete copulas and some subclasses.

### 3.4. Discrete t-norms and t-conorms

Among all subclasses of conjunctions, it is likely that the classes of discrete t-norms and discrete t-conorms have garnered the most significant interest within the research community. This is partially due to their remarkable versatility in tackling a multitude of practical problems, including applications in image processing [1, 2], and decision-making [3, 4] scenarios. We start with the definition of discrete t-norm.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Mayor et al. in [18] proposed a complete characterization of discrete copulas using permutation matrices, which are in fact a particular case of alternating sign matrices. The construction of a permutation matrix is derived from the permutation group of n elements, denoted as $\mathfrak{C}_n$.

**Definition 3.10.** ([53, Section 5.6]) A permutation matrix $P = (p_{i,j})$ is an $n \times n$ matrix such that there exists a permutation $\sigma \in \mathfrak{C}_n$ of $L_n \setminus \{0\}$ whereby $P$ is expressed as

$$p_{ij} = \begin{cases} 1, & \text{if } i = \sigma(j), \\ 0, & \text{otherwise,} \end{cases} \qquad (30)$$

for all $i, j \in \{1, \dots, n\}$.

They characterized discrete copulas using these matrices, yielding a result akin to Theorem 3.1. Subsequently, they were able to determine the cardinality of this class of discrete operators.

**Proposition 3.12.** ([18, Proposition 6]) Let $n \ge 1$, and let $K: L_n^2 \to L_n$ be a binary operator. Then, $K$ is a discrete copula if, and only if, there exists an $n \times n$ permutation matrix $P = (p_{ij})$ such that

$$K(x, y) = \begin{cases} 0, & \text{if } x = 0 \text{ or } y = 0, \\ \sum_{i=1}^{x} \sum_{j=1}^{y} p_{ij}, & \text{otherwise,} \end{cases} \qquad (31)$$

for all $x, y \in L_n$.

**Proposition 3.13.** ([18, Corollary 1]) Let $n \ge 1$, and let us denote by $\mathcal{K}_n$ the set of discrete copulas defined on $L_n$. Then,

$$|\mathcal{K}_n| = n!. \qquad (32)$$

The same characterization of Proposition 3.12 also applies to commutative discrete copulas, but taking symmetric permutation matrices as the equivalent object.

**Proposition 3.14.** ([18, Corollary 1]) Let $n \ge 1$, and let us denote by $\mathcal{K}_n^{\text{COM}}$ the set of commutative discrete copulas defined on $L_n$. Then,

$$|\mathcal{K}_n^{\text{COM}}| = \sum_{i=0}^{\lfloor n/2 \rfloor} \frac{n!}{2^i (n-2i)! i!} \qquad (33)$$

In addition to these two results, Mayor et al. [18] also studied other properties of discrete copulas, such as the associative and idempotent-free properties.

**Proposition 3.15.** Let $n \ge 1$, and let us denote by $\mathcal{K}_n^{\text{ASV}}$ and $\mathcal{K}_n^{\text{IDF}}$ the sets of associative discrete copulas and idempotent-free discrete copulas, respectively, defined on $L_n$. Then, the following statements hold:

(i) ([18, Corollary 4]) $|\mathcal{K}_n^{\text{ASV}}| = 2^{n-1}$,

(ii) ([18, Proposition 8]) The only associative and idempotent-free discrete copula is the Łukasiewicz copula, given by $K_L(x, y) = \max\{0, x + y - n\}$, for all $x, y \in L_n$. Therefore, $|\mathcal{K}_n^{\text{ASV}} \cap \mathcal{K}_n^{\text{IDF}}| = 1$.

For discrete quasi-copulas and discrete copulas, we illustrate the first 13 terms of the different enumerations in Table 4. Although the enumeration of $Q_n$ coincides with that of $C_n^{\text{SMT}}$, since they are the same class of operators and has been illustrated previously in Table 3, we show it again so that it can be compared with the enumerations of discrete copulas and some subclasses.

### 3.4. Discrete t-norms and t-conorms

Among all subclasses of conjunctions, it is likely that the classes of discrete t-norms and discrete t-conorms have garnered the most significant interest within the research community. This is partially due to their remarkable versatility in tackling a multitude of practical problems, including applications in image processing [1, 2], and decision-making [3, 4] scenarios. We start with the definition of discrete t-norm.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 4**</caption><thead><tr><th>n</th><th>Q<sub>n</sub></th><th>K<sub>n</sub></th><th>K<sup>COM</sup><sub>n</sub></th><th>K<sup>ASV</sup><sub>n</sub></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>7 = 7 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td></tr><tr><td>4</td><td>42 = 4.2 &middot; 10<sup>1</sup></td><td>24 = 2.4 &middot; 10<sup>1</sup></td><td>10 = 1 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>1</sup></td></tr><tr><td>5</td><td>429 = 4.29 &middot; 10<sup>2</sup></td><td>120 = 1.2 &middot; 10<sup>2</sup></td><td>26 = 2.6 &middot; 10<sup>1</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td></tr><tr><td>6</td><td>7,436 = 7.436 &middot; 10<sup>3</sup></td><td>720 = 7.2 &middot; 10<sup>2</sup></td><td>76 = 7.6 &middot; 10<sup>1</sup></td><td>32 = 3.2 &middot; 10<sup>1</sup></td></tr><tr><td>7</td><td>≈ 2.1835 &middot; 10<sup>5</sup></td><td>5,040 = 5.04 &middot; 10<sup>3</sup></td><td>232 = 2.32 &middot; 10<sup>2</sup></td><td>64 = 6.4 &middot; 10<sup>1</sup></td></tr><tr><td>8</td><td>≈ 1.085 &middot; 10<sup>7</sup></td><td>40,320 = 4.03210<sup>4</sup></td><td>764 = 7.64 &middot; 10<sup>2</sup></td><td>128 = 1.28 &middot; 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 9.1184 &middot; 10<sup>8</sup></td><td>3,628,800 = 3.6288 &middot; 10<sup>5</sup></td><td>2,620 = 2.62 &middot; 10<sup>3</sup></td><td>256 = 2.56 &middot; 10<sup>2</sup></td></tr><tr><td>10</td><td>≈ 1.2953 &middot; 10<sup>11</sup></td><td>≈ 3.6288 &middot; 10<sup>6</sup></td><td>9,496 = 9.496 &middot; 10<sup>3</sup></td><td>512 = 5.12 &middot; 10<sup>2</sup></td></tr><tr><td>11</td><td>≈ 3.1096 &middot; 10<sup>13</sup></td><td>≈ 3.9917 &middot; 10<sup>7</sup></td><td>35,696 = 3.5696 &middot; 10<sup>4</sup></td><td>1,024 = 1.024 &middot; 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 1.2611 &middot; 10<sup>16</sup></td><td>≈ 4.79 &middot; 10<sup>8</sup></td><td>≈ 1.4015 &middot; 10<sup>5</sup></td><td>2,048 = 2.048 &middot; 10<sup>3</sup></td></tr><tr><td>13</td><td>≈ 8.6394 &middot; 10<sup>18</sup></td><td>≈ 6.227 &middot; 10<sup>9</sup></td><td>≈ 5.685 &middot; 10<sup>5</sup></td><td>4,096 = 4.096 &middot; 10<sup>3</sup></td></tr></tbody></table>

Enumeration of discrete quasi-copulas and some classes of discrete copulas, for $n \le 13$. From left to right: discrete quasi-copulas, given by Proposition 3.9; discrete copulas in general, given by Proposition 3.13; commutative discrete copulas, given by Proposition 3.14; and finally associative discrete copulas, given by Proposition 3.15.

**Definition 3.11.** ([8, 22]) A discrete t-norm is a discrete conjunction $T : L_n^2 \to L_n$ which is commutative, associative and has $n$ as neutral element.

Let $\mathcal{T}_n$ denote the set of discrete t-norms on $L_n$. The problem of determining its in general goes back to De Baets and Mesiar [5]. They explored t-norms defined across diverse finite structures, which encompass finite chains among others. They used computational methods to generate all possible discrete t-norms defined on $L_n$ for $n \le 13$, and therefore they were able to determine $|\mathcal{T}_n|$ for these values of $n$. These values of $|\mathcal{T}_n|$ are listed on Table 5. Subsequent to this paper, no significant progress has been achieved for the cardinality of discrete t-norms in general; only some progress has been made for the enumeration of t-norms for some specific cases of bounded lattices (see [54, 55]). Nevertheless, certain subclasses of discrete t-norms have garnered attention, which we explain below.

Different properties have been studied for discrete t-norms. One of them is the fact of being Archimedean, whose original definition we do not include for the sake of clarity (refer to [8, Definition 7.2.10] and [22, Definition 2.5] for more information) but which is equivalent for discrete t-norms as being idempotent-free. The number of discrete t-norms that are Archimedean, $|\mathcal{T}_n^{ARC}|$, remains an open problem. Conversely, if a discrete t-norm $T$ is idempotent, it is necessarily the minimum discrete t-norm, i.e., $T(x, y) = T_M(x, y) = \min\{x, y\}$ for all $x, y \in L_n$, so if $\mathcal{T}_n^{IDE}$ represents the set of idempotent discrete t-norms, then $|\mathcal{T}_n^{IDE}| = 1$, for all $n \ge 1$. Particular attention is directed toward the category of smooth discrete t-norms. These were originally characterized by Mayor and Torrens in [56, Theorem 6] through the use of the ordinal sum concept, albeit with notation and terminology that diverged from present conventions. Subsequently, in [8, Theorem 7.2.16], the same authors revisited this result, presenting it using a more present-day notation.

**Theorem 3.2.** ([8, Proposition 7.3.8]) Let $T: L_n^2 \to L_n$ be a t-norm on $L_n$. Then, $T$ is smooth if, and only if, there exists a natural number $r$ with $0 \le r \le n-1$ and a subset

$$I = \{0 = a_0 < a_1 < \dots < a_r < a_{r+1} = n\} \subseteq L_n,$$

such that $T$ is given by

$$T(x, y) = \begin{cases} \max\{a_i, x + y - a_{i+1}\}, & \text{if } (x, y) \in [a_i, a_{i+1}]^2 \text{ for some } 0 \le i \le r, \\ \min\{x, y\}, & \text{otherwise,} \end{cases} \quad (34)$$

for all $x, y \in L_n$.

This result implies that the expression of any smooth discrete t-norm $T$ is uniquely determined by the set $I$. Since as many sets can be constructed as there are elements in the power set of the interval
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 4**</caption><thead><tr><th>n</th><th>Q<sub>n</sub></th><th>K<sub>n</sub></th><th>K<sup>COM</sup><sub>n</sub></th><th>K<sup>ASV</sup><sub>n</sub></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>7 = 7 &middot; 10<sup>0</sup></td><td>6 = 6 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td></tr><tr><td>4</td><td>42 = 4.2 &middot; 10<sup>1</sup></td><td>24 = 2.4 &middot; 10<sup>1</sup></td><td>10 = 1 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>1</sup></td></tr><tr><td>5</td><td>429 = 4.29 &middot; 10<sup>2</sup></td><td>120 = 1.2 &middot; 10<sup>2</sup></td><td>26 = 2.6 &middot; 10<sup>1</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td></tr><tr><td>6</td><td>7,436 = 7.436 &middot; 10<sup>3</sup></td><td>720 = 7.2 &middot; 10<sup>2</sup></td><td>76 = 7.6 &middot; 10<sup>1</sup></td><td>32 = 3.2 &middot; 10<sup>1</sup></td></tr><tr><td>7</td><td>≈ 2.1835 &middot; 10<sup>5</sup></td><td>5,040 = 5.04 &middot; 10<sup>3</sup></td><td>232 = 2.32 &middot; 10<sup>2</sup></td><td>64 = 6.4 &middot; 10<sup>1</sup></td></tr><tr><td>8</td><td>≈ 1.085 &middot; 10<sup>7</sup></td><td>40,320 = 4.03210<sup>4</sup></td><td>764 = 7.64 &middot; 10<sup>2</sup></td><td>128 = 1.28 &middot; 10<sup>2</sup></td></tr><tr><td>9</td><td>≈ 9.1184 &middot; 10<sup>8</sup></td><td>3,628,800 = 3.6288 &middot; 10<sup>5</sup></td><td>2,620 = 2.62 &middot; 10<sup>3</sup></td><td>256 = 2.56 &middot; 10<sup>2</sup></td></tr><tr><td>10</td><td>≈ 1.2953 &middot; 10<sup>11</sup></td><td>≈ 3.6288 &middot; 10<sup>6</sup></td><td>9,496 = 9.496 &middot; 10<sup>3</sup></td><td>512 = 5.12 &middot; 10<sup>2</sup></td></tr><tr><td>11</td><td>≈ 3.1096 &middot; 10<sup>13</sup></td><td>≈ 3.9917 &middot; 10<sup>7</sup></td><td>35,696 = 3.5696 &middot; 10<sup>4</sup></td><td>1,024 = 1.024 &middot; 10<sup>3</sup></td></tr><tr><td>12</td><td>≈ 1.2611 &middot; 10<sup>16</sup></td><td>≈ 4.79 &middot; 10<sup>8</sup></td><td>≈ 1.4015 &middot; 10<sup>5</sup></td><td>2,048 = 2.048 &middot; 10<sup>3</sup></td></tr><tr><td>13</td><td>≈ 8.6394 &middot; 10<sup>18</sup></td><td>≈ 6.227 &middot; 10<sup>9</sup></td><td>≈ 5.685 &middot; 10<sup>5</sup></td><td>4,096 = 4.096 &middot; 10<sup>3</sup></td></tr></tbody></table>

Enumeration of discrete quasi-copulas and some classes of discrete copulas, for $n \le 13$. From left to right: discrete quasi-copulas, given by Proposition 3.9; discrete copulas in general, given by Proposition 3.13; commutative discrete copulas, given by Proposition 3.14; and finally associative discrete copulas, given by Proposition 3.15.

**Definition 3.11.** ([8, 22]) A discrete t-norm is a discrete conjunction $T : L_n^2 \to L_n$ which is commutative, associative and has $n$ as neutral element.

Let $\mathcal{T}_n$ denote the set of discrete t-norms on $L_n$. The problem of determining its in general goes back to De Baets and Mesiar [5]. They explored t-norms defined across diverse finite structures, which encompass finite chains among others. They used computational methods to generate all possible discrete t-norms defined on $L_n$ for $n \le 13$, and therefore they were able to determine $|\mathcal{T}_n|$ for these values of $n$. These values of $|\mathcal{T}_n|$ are listed on Table 5. Subsequent to this paper, no significant progress has been achieved for the cardinality of discrete t-norms in general; only some progress has been made for the enumeration of t-norms for some specific cases of bounded lattices (see [54, 55]). Nevertheless, certain subclasses of discrete t-norms have garnered attention, which we explain below.

Different properties have been studied for discrete t-norms. One of them is the fact of being Archimedean, whose original definition we do not include for the sake of clarity (refer to [8, Definition 7.2.10] and [22, Definition 2.5] for more information) but which is equivalent for discrete t-norms as being idempotent-free. The number of discrete t-norms that are Archimedean, $|\mathcal{T}_n^{ARC}|$, remains an open problem. Conversely, if a discrete t-norm $T$ is idempotent, it is necessarily the minimum discrete t-norm, i.e., $T(x, y) = T_M(x, y) = \min\{x, y\}$ for all $x, y \in L_n$, so if $\mathcal{T}_n^{IDE}$ represents the set of idempotent discrete t-norms, then $|\mathcal{T}_n^{IDE}| = 1$, for all $n \ge 1$. Particular attention is directed toward the category of smooth discrete t-norms. These were originally characterized by Mayor and Torrens in [56, Theorem 6] through the use of the ordinal sum concept, albeit with notation and terminology that diverged from present conventions. Subsequently, in [8, Theorem 7.2.16], the same authors revisited this result, presenting it using a more present-day notation.

**Theorem 3.2.** ([8, Proposition 7.3.8]) Let $T: L_n^2 \to L_n$ be a t-norm on $L_n$. Then, $T$ is smooth if, and only if, there exists a natural number $r$ with $0 \le r \le n-1$ and a subset

$$I = \{0 = a_0 < a_1 < \dots < a_r < a_{r+1} = n\} \subseteq L_n,$$

such that $T$ is given by

$$T(x, y) = \begin{cases} \max\{a_i, x + y - a_{i+1}\}, & \text{if } (x, y) \in [a_i, a_{i+1}]^2 \text{ for some } 0 \le i \le r, \\ \min\{x, y\}, & \text{otherwise,} \end{cases} \quad (34)$$

for all $x, y \in L_n$.

This result implies that the expression of any smooth discrete t-norm $T$ is uniquely determined by the set $I$. Since as many sets can be constructed as there are elements in the power set of the interval
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

[1, n - 1], Proposition 3.16 establishes its cardinality. But there is a further consequence of Theorem 3.2. When we simultaneously impose the conditions of being smooth and Archimedean, and bearing in mind that each component $\max\{a_i, x + y - a_{i+1}\}$ of Equation (34) is already idempotent-free on each interval $[a_i, a_{i+1}]$, the sole viable choice is $I = \{0, n\}$. This leads to the specific expression of $T$ as $T(x, y) = T_L(x, y) = \max\{0, x + y - n\}$, for all $x, y \in L_n$, which is renowned as Łukasiewicz discrete t-norm. Therefore, it can be inferred that for a given $n \ge 1$ there exists only one smooth and Archimedean discrete t-norm.

**Proposition 3.16.** Let $n \ge 1$, and let us denote by $\mathcal{T}_n^{\text{SMT}}$ and $\mathcal{T}_n^{\text{ARC}}$ the sets of smooth discrete t-norms and Archimedean discrete t-norms, respectively, defined on $L_n$. Then,

$$ (i) \quad ([8, \text{Proposition 7.3.7}]) \quad |\mathcal{T}_n^{\text{SMT}} \cap \mathcal{T}_n^{\text{ARC}}| = 1. $$

$$ (ii) \quad ([8, \text{Corollary 7.3.12}]) \quad |\mathcal{T}_n^{\text{SMT}}| = 2^{n-1}. $$

Another class of operators recently proposed by Bejines and Navara [12] is that of maximal Archimedean discrete t-norms.

**Definition 3.12.** ([12, Definition 2.4]) A discrete t-norm $T$ is maximal Archimedean if it is Archimedean and there is no other Archimedean discrete t-norm $T'$ on $L_n$ satisfying $T(x, y) \le T'(x, y)$, for all $x, y \in L_n$.

If we denote by $\mathcal{T}_n^{\text{MACH}}$ the set of maximal Archimedean discrete t-norms on $L_n$, in [12] it is shown that its cardinality is given by the Fibonacci sequence. In fact, they expressed every discrete t-norm of $\mathcal{T}_n^{\text{MACH}}$ as the extension of a discrete t-norm of $\mathcal{T}_{n-1}^{\text{MACH}}$ or $\mathcal{T}_{n-2}^{\text{MACH}}$, so they ended up with this recursive formula. We omit the details of this decomposition due to its length and because it involves a large number of previous technical results.

**Proposition 3.17.** ([12, Section 4.4]) Let $n \ge 1$, and let us denote by $\mathcal{T}_n^{\text{MACH}}$ the set of maximal Archimedean discrete t-norms defined on $L_n$. Then, the number of maximal Archimedean discrete t-norms satisfies the recurrence equation

$$ |\mathcal{T}_n^{\text{MACH}}| = |\mathcal{T}_{n-1}^{\text{MACH}}| + |\mathcal{T}_{n-2}^{\text{MACH}}|, \quad (35) $$

for all $n \ge 3$, with $|\mathcal{T}_1^{\text{MACH}}| = |\mathcal{T}_2^{\text{MACH}}| = 1$. Therefore, $|\mathcal{T}_n^{\text{MACH}}|$ is the n-th element of the Fibonacci sequence, and its general formula is given by

$$ |\mathcal{T}_n^{\text{MACH}}| = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}. \quad (36) $$

As a particular case of discrete disjunction, we have the concept of discrete t-conorm, which is the other class of operators we recall in this section.

**Definition 3.13.** ([8, 22]) A discrete t-conorm is a discrete disjunction $S: L_n^2 \to L_n$ which is commutative, associative and has 0 as neutral element.

Discrete t-norms and t-conorms have very similar definitions, with the position of the neutral element being the only difference. Since the origin of these operators, the duality between discrete t-norms and discrete t-conorms is already known and, in fact, served as the inspiration for the duality between discrete conjunctions and disjunctions presented in Proposition 3.11. As a consequence, cardinalities between some classes of discrete t-norms and t-conorms are also preserved.

**Proposition 3.18.** ([8, 22]) Let $n \ge 1$, and let us denote by $\mathcal{T}_n$ and $S_n$ the sets of discrete t-norms and discrete t-conorms, respectively, defined on $L_n$. Let $\varphi: \mathcal{T}_n \to S_n$ be defined point-wise by

$$ \varphi(T)(x, y) = n - T(n - x, n - y), $$

for all $x, y \in L_n$, where $T \in \mathcal{T}_n$. Then, $\varphi(T)$ is a discrete t-conorm, and the transformation $\varphi$ is a bijection.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

[1, n - 1], Proposition 3.16 establishes its cardinality. But there is a further consequence of Theorem 3.2. When we simultaneously impose the conditions of being smooth and Archimedean, and bearing in mind that each component max {$a_i, x + y - a_{i+1}$} of Equation (34) is already idempotent-free on each interval $[a_i, a_{i+1}]$, the sole viable choice is $I = \{0, n\}$. This leads to the specific expression of $T$ as $T(x, y) = T_L(x, y) = \max \{0, x + y - n\}$, for all $x, y \in L_n$, which is renowned as Łukasiewicz discrete t-norm. Therefore, it can be inferred that for a given $n \ge 1$ there exists only one smooth and Archimedean discrete t-norm.

**Proposition 3.16.** Let $n \ge 1$, and let us denote by $\mathcal{T}_n^{\text{SMT}}$ and $\mathcal{T}_n^{\text{ARC}}$ the sets of smooth discrete t-norms and Archimedean discrete t-norms, respectively, defined on $L_n$. Then,

$$ (i) \quad ([8, \text{Proposition 7.3.7}]) \quad |\mathcal{T}_n^{\text{SMT}} \cap \mathcal{T}_n^{\text{ARC}}| = 1. $$

$$ (ii) \quad ([8, \text{Corollary 7.3.12}]) \quad |\mathcal{T}_n^{\text{SMT}}| = 2^{n-1}. $$

Another class of operators recently proposed by Bejines and Navara [12] is that of maximal Archimedean discrete t-norms.

**Definition 3.12.** ([12, Definition 2.4]) A discrete t-norm $T$ is maximal Archimedean if it is Archimedean and there is no other Archimedean discrete t-norm $T'$ on $L_n$ satisfying $T(x, y) \le T'(x, y)$, for all $x, y \in L_n$.

If we denote by $\mathcal{T}_n^{\text{MACH}}$ the set of maximal Archimedean discrete t-norms on $L_n$, in [12] it is shown that its cardinality is given by the Fibonacci sequence. In fact, they expressed every discrete t-norm of $\mathcal{T}_n^{\text{MACH}}$ as the extension of a discrete t-norm of $\mathcal{T}_{n-1}^{\text{MACH}}$ or $\mathcal{T}_{n-2}^{\text{MACH}}$, so they ended up with this recursive formula. We omit the details of this decomposition due to its length and because it involves a large number of previous technical results.

**Proposition 3.17.** ([12, Section 4.4]) Let $n \ge 1$, and let us denote by $\mathcal{T}_n^{\text{MACH}}$ the set of maximal Archimedean discrete t-norms defined on $L_n$. Then, the number of maximal Archimedean discrete t-norms satisfies the recurrence equation

$$ |\mathcal{T}_n^{\text{MACH}}| = |\mathcal{T}_{n-1}^{\text{MACH}}| + |\mathcal{T}_{n-2}^{\text{MACH}}|, \quad (35) $$

for all $n \ge 3$, with $|\mathcal{T}_1^{\text{MACH}}| = |\mathcal{T}_2^{\text{MACH}}| = 1$. Therefore, $|\mathcal{T}_n^{\text{MACH}}|$ is the n-th element of the Fibonacci sequence, and its general formula is given by

$$ |\mathcal{T}_n^{\text{MACH}}| = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}. \quad (36) $$

As a particular case of discrete disjunction, we have the concept of discrete t-conorm, which is the other class of operators we recall in this section.

**Definition 3.13.** ([8, 22]) A discrete t-conorm is a discrete disjunction $S: L_n^2 \to L_n$ which is commutative, associative and has 0 as neutral element.

Discrete t-norms and t-conorms have very similar definitions, with the position of the neutral element being the only difference. Since the origin of these operators, the duality between discrete t-norms and discrete t-conorms is already known and, in fact, served as the inspiration for the duality between discrete conjunctions and disjunctions presented in Proposition 3.11. As a consequence, cardinalities between some classes of discrete t-norms and t-conorms are also preserved.

**Proposition 3.18.** ([8, 22]) Let $n \ge 1$, and let us denote by $\mathcal{T}_n$ and $S_n$ the sets of discrete t-norms and discrete t-conorms, respectively, defined on $L_n$. Let $\varphi: \mathcal{T}_n \to S_n$ be defined point-wise by

$$ \varphi(T)(x, y) = n - T(n - x, n - y), $$

for all $x, y \in L_n$, where $T \in \mathcal{T}_n$. Then, $\varphi(T)$ is a discrete t-conorm, and the transformation $\varphi$ is a bijection.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 5**</caption><thead><tr><th>n</th><th>|T<sub>n</sub>|, S<sub>n</sub></th><th>|T<sub>n</sub><sup>SMT</sup>|, S<sub>n</sub><sup>SMT</sup></th><th>|T<sub>n</sub><sup>MACH</sup>|, S<sub>n</sub><sup>MACH</sup></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>2</td><td>2 = 2 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td><td>1 = 1 &middot; 10<sup>0</sup></td></tr><tr><td>3</td><td>6 = 6 &middot; 10<sup>0</sup></td><td>4 = 4 &middot; 10<sup>0</sup></td><td>2 = 2 &middot; 10<sup>0</sup></td></tr><tr><td>4</td><td>22 = 2.2 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>0</sup></td><td>3 = 3 &middot; 10<sup>0</sup></td></tr><tr><td>5</td><td>94 = 9.4 &middot; 10<sup>1</sup></td><td>16 = 1.6 &middot; 10<sup>1</sup></td><td>5 = 5 &middot; 10<sup>0</sup></td></tr><tr><td>6</td><td>451 = 4.51 &middot; 10<sup>2</sup></td><td>32 = 3.2 &middot; 10<sup>1</sup></td><td>8 = 8 &middot; 10<sup>0</sup></td></tr><tr><td>7</td><td>2,386 = 2.386 &middot; 10<sup>3</sup></td><td>64 = 6.4 &middot; 10<sup>1</sup></td><td>13 = 1.3 &middot; 10<sup>1</sup></td></tr><tr><td>8</td><td>13,775 = 1.3775 &middot; 10<sup>4</sup></td><td>128 = 1.28 &middot; 10<sup>2</sup></td><td>21 = 2.1 &middot; 10<sup>1</sup></td></tr><tr><td>9</td><td>86,417 = 8.6417 &middot; 10<sup>4</sup></td><td>256 = 2.56 &middot; 10<sup>2</sup></td><td>34 = 3.4 &middot; 10<sup>1</sup></td></tr><tr><td>10</td><td>590,489 &asymp; 5.9049 &middot; 10<sup>5</sup></td><td>512 = 5.12 &middot; 10<sup>2</sup></td><td>55 = 5.5 &middot; 10<sup>1</sup></td></tr><tr><td>11</td><td>4,446,029 &asymp; 4.446 &middot; 10<sup>6</sup></td><td>1,024 = 1.024 &middot; 10<sup>3</sup></td><td>89 = 8.9 &middot; 10<sup>1</sup></td></tr><tr><td>12</td><td>37,869,449 &asymp; 3.7869 &middot; 10<sup>7</sup></td><td>2,048 = 2.048 &middot; 10<sup>3</sup></td><td>144 = 1.44 &middot; 10<sup>2</sup></td></tr><tr><td>13</td><td>382,549,464 &asymp; 3.8255 &middot; 10<sup>8</sup></td><td>4,096 = 4.096 &middot; 10<sup>3</sup></td><td>233 = 2.33 &middot; 10<sup>2</sup></td></tr></tbody></table>

Enumeration of some classes of discrete t-norms and t-conorms, for $n \le 13$. From left to right: discrete t-norms and t-conorms in general, obtained computationally; smooth discrete t-norms and t-conorms, given by Proposition 3.16; and finally maximal Archimedean discrete t-norms and minimal Archimedean discrete t-conorms, given by Proposition 3.17.

So, with this result, the enumeration of discrete t-conorms coincides with that of discrete t-norms. However, since $|T_n|$ lacks a general term, the same applies to $|S_n|$. Also for the Archimedean property for discrete t-conorms, which is equivalent to $S(x, x) > x$ for all $x \in L_n \setminus \{0, n\}$: the enumeration of $|S_n^{ARC}|$ coincides with the enumeration of $|T_n^{ARC}|$, but a general formula for the enumeration of these operators is still unknown.

Regarding the other formulas, they can also be adapted for discrete t-conorms. With the duality between discrete t-norms and t-conorms of Proposition 3.18, a discrete t-norm is smooth if and only if its dual discrete t-conorm is smooth. Also, there exists a unique smooth and Archimedean discrete t-conorm, which is the Łukasiewicz discrete t-conorm $S_L(x, y) = \min\{n, x + y\}$, for all $x, y \in L_n$. Finally, for maximal Archimedean discrete t-norms, there is no precedent in the existing literature regarding this concept for discrete t-conorms. However, it is clear that Definition 3.12 can be adjusted by replacing a discrete t-norm $T$ by a discrete t-conorm $S$, keeping the Archimedean property and replacing the property of being maximal with that of being minimal, i.e., when there is not other Archimedean discrete t-conorm $S'$ such that $S'(x, y) \le S(x, y)$, giving rise to the minimal Archimedean discrete t-conorm. With this discussion, we condense all these formulas into the following three items.

* Smooth discrete t-conorms, $S_n^{SMT}$. Its cardinality coincides with $|T_n^{SMT}|$, given in Proposition 3.16.

* Smooth and Archimedean discrete t-conorms, $S_n^{SMT} \cap S_n^{ARC}$. Its cardinality coincides with $|T_n^{SMT} \cap T_n^{ARC}|$, given in Proposition 3.16.

* Minimal Archimedean discrete t-conorms, $S_n^{MACH}$. Its cardinality coincides with $|T_n^{MACH}|$, given in Proposition 3.17.

All these classes are listed in Table 5 for $n \le 13$.

### 3.5. Discrete uninorms

To generalize the concepts of discrete t-norms and discrete t-conorms, the concept of discrete uninorm appears, which was introduced on discrete settings for the first time in [7]. Discrete uninorms were introduced based on the observation that discrete t-norms and discrete t-conorms primarily differ in the value of their neutral element, and therefore are particular cases of commutative and associative discrete aggregation functions, whose neutral element is any $e \in L_n$. In [29] there is a comprehensive survey about uninorms on [0, 1] and on $L_n$ and in [22] there is a section dedicated solely addressing discrete uninorms.

**Definition 3.14.** ([7, 29]) A discrete uninorm is a discrete aggregation function $U: L_n^2 \to L_n$, which is associative, commutative and has neutral element $e \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 5**</caption><thead><tr><th>n</th><th>|T<sub>n</sub>|, S<sub>n</sub></th><th>|T<sub>n</sub><sup>SMT</sup>|, S<sub>n</sub><sup>SMT</sup></th><th>|T<sub>n</sub><sup>MACH</sup>|, S<sub>n</sub><sup>MACH</sup></th></tr></thead><tbody><tr><td>1</td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td></tr><tr><td>2</td><td>2 = 2 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td><td>1 = 1 · 10<sup>0</sup></td></tr><tr><td>3</td><td>6 = 6 · 10<sup>0</sup></td><td>4 = 4 · 10<sup>0</sup></td><td>2 = 2 · 10<sup>0</sup></td></tr><tr><td>4</td><td>22 = 2.2 · 10<sup>1</sup></td><td>8 = 8 · 10<sup>0</sup></td><td>3 = 3 · 10<sup>0</sup></td></tr><tr><td>5</td><td>94 = 9.4 · 10<sup>1</sup></td><td>16 = 1.6 · 10<sup>1</sup></td><td>5 = 5 · 10<sup>0</sup></td></tr><tr><td>6</td><td>451 = 4.51 · 10<sup>2</sup></td><td>32 = 3.2 · 10<sup>1</sup></td><td>8 = 8 · 10<sup>0</sup></td></tr><tr><td>7</td><td>2,386 = 2.386 · 10<sup>3</sup></td><td>64 = 6.4 · 10<sup>1</sup></td><td>13 = 1.3 · 10<sup>1</sup></td></tr><tr><td>8</td><td>13,775 = 1.3775 · 10<sup>4</sup></td><td>128 = 1.28 · 10<sup>2</sup></td><td>21 = 2.1 · 10<sup>1</sup></td></tr><tr><td>9</td><td>86,417 = 8.6417 · 10<sup>4</sup></td><td>256 = 2.56 · 10<sup>2</sup></td><td>34 = 3.4 · 10<sup>1</sup></td></tr><tr><td>10</td><td>590,489 ≈ 5.9049 · 10<sup>5</sup></td><td>512 = 5.12 · 10<sup>2</sup></td><td>55 = 5.5 · 10<sup>1</sup></td></tr><tr><td>11</td><td>4,446,029 ≈ 4.446 · 10<sup>6</sup></td><td>1,024 = 1.024 · 10<sup>3</sup></td><td>89 = 8.9 · 10<sup>1</sup></td></tr><tr><td>12</td><td>37,869,449 ≈ 3.7869 · 10<sup>7</sup></td><td>2,048 = 2.048 · 10<sup>3</sup></td><td>144 = 1.44 · 10<sup>2</sup></td></tr><tr><td>13</td><td>382,549,464 ≈ 3.8255 · 10<sup>8</sup></td><td>4,096 = 4.096 · 10<sup>3</sup></td><td>233 = 2.33 · 10<sup>2</sup></td></tr></tbody></table>

Enumeration of some classes of discrete t-norms and t-conorms, for $n \le 13$. From left to right: discrete t-norms and t-conorms in general, obtained computationally; smooth discrete t-norms and t-conorms, given by Proposition 3.16; and finally maximal Archimedean discrete t-norms and minimal Archimedean discrete t-conorms, given by Proposition 3.17.

So, with this result, the enumeration of discrete t-conorms coincides with that of discrete t-norms. However, since $|T_n|$ lacks a general term, the same applies to $|S_n|$. Also for the Archimedean property for discrete t-conorms, which is equivalent to $S(x, x) > x$ for all $x \in L_n \setminus \{0, n\}$: the enumeration of $|S_n^{ARC}|$ coincides with the enumeration of $|T_n^{ARC}|$, but a general formula for the enumeration of these operators is still unknown.

Regarding the other formulas, they can also be adapted for discrete t-conorms. With the duality between discrete t-norms and t-conorms of Proposition 3.18, a discrete t-norm is smooth if and only if its dual discrete t-conorm is smooth. Also, there exists a unique smooth and Archimedean discrete t-conorm, which is the Łukasiewicz discrete t-conorm $S_L(x, y) = \min\{n, x + y\}$, for all $x, y \in L_n$. Finally, for maximal Archimedean discrete t-norms, there is no precedent in the existing literature regarding this concept for discrete t-conorms. However, it is clear that Definition 3.12 can be adjusted by replacing a discrete t-norm $T$ by a discrete t-conorm $S$, keeping the Archimedean property and replacing the property of being maximal with that of being minimal, i.e., when there is not other Archimedean discrete t-conorm $S'$ such that $S'(x, y) \le S(x, y)$, giving rise to the minimal Archimedean discrete t-conorm. With this discussion, we condense all these formulas into the following three items.

* Smooth discrete t-conorms, $S_n^{SMT}$. Its cardinality coincides with $|T_n^{SMT}|$, given in Proposition 3.16.

* Smooth and Archimedean discrete t-conorms, $S_n^{SMT} \cap S_n^{ARC}$. Its cardinality coincides with $|T_n^{SMT} \cap T_n^{ARC}|$, given in Proposition 3.16.

* Minimal Archimedean discrete t-conorms, $S_n^{MACH}$. Its cardinality coincides with $|T_n^{MACH}|$, given in Proposition 3.17.

All these classes are listed in Table 5 for $n \le 13$.

### 3.5. Discrete uninorms

To generalize the concepts of discrete t-norms and discrete t-conorms, the concept of discrete uninorm appears, which was introduced on discrete settings for the first time in [7]. Discrete uninorms were introduced based on the observation that discrete t-norms and discrete t-conorms primarily differ in the value of their neutral element, and therefore are particular cases of commutative and associative discrete aggregation functions, whose neutral element is any $e \in L_n$. In [29] there is a comprehensive survey about uninorms on [0, 1] and on $L_n$ and in [22] there is a section dedicated solely addressing discrete uninorms.

**Definition 3.14.** ([7, 29]) A discrete uninorm is a discrete aggregation function $U : L_n^2 \to L_n$, which is associative, commutative and has neutral element $e \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Note that a discrete uninorm becomes a discrete t-conorm whenever $e = 0$, while it is a discrete t-norm whenever $e = n$. When $0 < e < n$, it is termed a proper discrete uninorm. In the latter case, it can be described as follows: its restriction on the region $[0, e]^2$ is a discrete t-norm $T_U$ on $[0, e]$, whereas its restriction to $[e, n]^2$ is a discrete t-conorm $S_U$ on $[e, n]$. We use the notation $U = \langle T_U, e, S_U \rangle$ to represent this behavior. In the remaining domain, denoted by

$$C_e = [0, e-1] \times [e+1, n] \cup [e+1, n] \times [0, e-1], \qquad (37)$$

and referred to as the compensation space, it holds that $\min\{x, y\} \le U(x, y) \le \max\{x, y\}$, for all $(x, y) \in C_e$. This inequality allows us to obtain that $U(0, n) \in \{0, n\}$, and discrete uninorms such that $U(0, n) = 0$ are called conjunctive, whereas those with $U(0, n) = n$ are called disjunctive. Since the general behavior of a uninorm depends on the nature of its underlying t-norm and t-conorm, and on the value it takes in the region $C_e$, several subclasses of uninorms have appeared in the literature according to the behavior of each of these components. As an observation, we note that on $L_n$ there are no smooth discrete uninorms (see [7, Corollary 4.4]), just as there are no continuous uninorms on $[0, 1]$ (see [29]).

As mentioned in Section 3.4, since a general formula for the number of discrete t-norms and discrete t-conorms is not known, it is foreseeable that the number of discrete uninorms is also not known given that its structure is a combination of a discrete t-norm and discrete t-conorm. Indeed, this issue remains an open problem, and to the best of our knowledge there are no references in the literature that have tackled it. Still, a formula can be conjectured for the cardinality of $U_n^{\text{MIN}}$, the set of discrete uninorms satisfying $U(x, y) = \min\{x, y\}$ for all $(x, y) \in C_e$, and $U_n^{\text{MAX}}$, the set of discrete uninorms satisfying $U(x, y) = \max\{x, y\}$ for all $(x, y) \in C_e$.

**Remark 3.5.** Although the number of discrete t-norms and t-conorms is unknown in general, a formula for $|U_n^{\text{MIN}}|$ and $|U_n^{\text{MAX}}|$ can be proposed, since these subclasses of discrete uninorms cannot take any expression in $C_e$, only the minimum or maximum. A first formula can be

$$|U_n^{\text{MIN}}| = |U_n^{\text{MAX}}| = \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |S_{n-e}|,$$

for all $n \ge 2$. However, bearing in mind that $|S_n| = |\mathcal{T}_n|$, it can be slightly simplified as

$$|U_n^{\text{MIN}}| = |U_n^{\text{MAX}}| = \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |\mathcal{T}_{n-e}|,$$

for all $n \ge 2$. Let us note that in these expressions we have that $e \in \{1, \dots, n-1\}$, since in order to define the discrete uninorms belonging to $U_n^{\text{MIN}}$ and $U_n^{\text{MAX}}$ the set $C_e$ must be considered, and to be it well defined the extreme values $e = 0$ and $e = n$ must be excluded.

Of the classes of discrete uninorms for which a formula for their cardinality is known, let us start with idempotent discrete uninorms, that have been characterized from three distinct perspectives, but only two focus on their enumeration. The first technique can be found in [10], where De Baets et al. propose a characterization grounded in decreasing unary functions.

**Theorem 3.3.** ([10, Theorem 3]) A binary operation $U$ on $L_n$ with neutral element $0 < e < n$ is an idempotent uninorm on $L_n$ if, and only if, there exists a decreasing function $g : [0, e] \to [e, n]$ with $g(e) = e$ such that

$$U(x, y) = \begin{cases} \min\{x, y\}, & \text{if } y \le \bar{g}(x) \text{ and } x \le \bar{g}(0), \\ \max\{x, y\}, & \text{otherwise,} \end{cases} \qquad (38)$$

for all $x, y \in L_n$, where $\bar{g} : L_n \to L_n$ is given by

$$\bar{g}(x) = \begin{cases} g(x), & \text{if } x \le e, \\ \max\{z \in [0, e] \mid g(z) \ge x\}, & \text{if } e \le x \le g(0), \\ 0, & \text{if } x > g(0), \end{cases} \qquad (39)$$

for all $x \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Note that a discrete uninorm becomes a discrete t-conorm whenever $e = 0$, while it is a discrete t-norm whenever $e = n$. When $0 < e < n$, it is termed a proper discrete uninorm. In the latter case, it can be described as follows: its restriction on the region $[0, e]^2$ is a discrete t-norm $T_U$ on $[0, e]$, whereas its restriction to $[e, n]^2$ is a discrete t-conorm $S_U$ on $[e, n]$. We use the notation $U = \langle T_U, e, S_U \rangle$ to represent this behavior. In the remaining domain, denoted by

$$C_e = [0, e-1] \times [e+1, n] \cup [e+1, n] \times [0, e-1], \quad (37)$$

and referred to as the compensation space, it holds that $\min\{x, y\} \le U(x, y) \le \max\{x, y\}$, for all $(x, y) \in C_e$. This inequality allows us to obtain that $U(0, n) \in \{0, n\}$, and discrete uninorms such that $U(0, n) = 0$ are called conjunctive, whereas those with $U(0, n) = n$ are called disjunctive. Since the general behavior of a uninorm depends on the nature of its underlying t-norm and t-conorm, and on the value it takes in the region $C_e$, several subclasses of uninorms have appeared in the literature according to the behavior of each of these components. As an observation, we note that on $L_n$ there are no smooth discrete uninorms (see [7, Corollary 4.4]), just as there are no continuous uninorms on $[0, 1]$ (see [29]).

As mentioned in Section 3.4, since a general formula for the number of discrete t-norms and discrete t-conorms is not known, it is foreseeable that the number of discrete uninorms is also not known given that its structure is a combination of a discrete t-norm and discrete t-conorm. Indeed, this issue remains an open problem, and to the best of our knowledge there are no references in the literature that have tackled it. Still, a formula can be conjectured for the cardinality of $U_n^{\text{MIN}}$, the set of discrete uninorms satisfying $U(x, y) = \min\{x, y\}$ for all $(x, y) \in C_e$, and $U_n^{\text{MAX}}$, the set of discrete uninorms satisfying $U(x, y) = \max\{x, y\}$ for all $(x, y) \in C_e$.

**Remark 3.5.** Although the number of discrete t-norms and t-conorms is unknown in general, a formula for $|U_n^{\text{MIN}}|$ and $|U_n^{\text{MAX}}|$ can be proposed, since these subclasses of discrete uninorms cannot take any expression in $C_e$, only the minimum or maximum. A first formula can be

$$|U_n^{\text{MIN}}| = |U_n^{\text{MAX}}| = \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |S_{n-e}|,$$

for all $n \ge 2$. However, bearing in mind that $|S_n| = |\mathcal{T}_n|$, it can be slightly simplified as

$$|U_n^{\text{MIN}}| = |U_n^{\text{MAX}}| = \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |\mathcal{T}_{n-e}|,$$

for all $n \ge 2$. Let us note that in these expressions we have that $e \in \{1, \dots, n-1\}$, since in order to define the discrete uninorms belonging to $U_n^{\text{MIN}}$ and $U_n^{\text{MAX}}$ the set $C_e$ must be considered, and to be it well defined the extreme values $e = 0$ and $e = n$ must be excluded.

Of the classes of discrete uninorms for which a formula for their cardinality is known, let us start with idempotent discrete uninorms, that have been characterized from three distinct perspectives, but only two focus on their enumeration. The first technique can be found in [10], where De Baets et al. propose a characterization grounded in decreasing unary functions.

**Theorem 3.3.** ([10, Theorem 3]) A binary operation $U$ on $L_n$ with neutral element $0 < e < n$ is an idempotent uninorm on $L_n$ if, and only if, there exists a decreasing function $g : [0, e] \to [e, n]$ with $g(e) = e$ such that

$$U(x, y) = \begin{cases} \min\{x, y\}, & \text{if } y \le \bar{g}(x) \text{ and } x \le \bar{g}(0), \\ \max\{x, y\}, & \text{otherwise,} \end{cases} \quad (38)$$

for all $x, y \in L_n$, where $\bar{g} : L_n \to L_n$ is given by

$$\bar{g}(x) = \begin{cases} g(x), & \text{if } x \le e, \\ \max\{z \in [0, e] \mid g(z) \ge x\}, & \text{if } e \le x \le g(0), \\ 0, & \text{if } x > g(0), \end{cases} \quad (39)$$

for all $x \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

This result enabled the authors of [10] to determine the number of idempotent discrete uninorms with neutral element $e \in L_n \setminus \{0, n\}$ in particular, and also the number of idempotent discrete uninorms in general. Observe that in this result they convert the enumeration of these binary operators into a count of decreasing unary operators with fixed point $e$, whose enumeration is simpler. This enumeration can be found in Theorem 3.5.

The second approach is presented by Couceiro et al. in [14, 16]. In these papers, the authors reapply the methodology used in the study for the enumeration of several families of discrete aggregation functions described in Propositions 3.3 and 3.4, again from the perspective of the level curves of binary operators and their geometry. For that, they introduced a powerful tool, namely, the concept of the F-degree sequence of a binary operator $F: L_n^2 \to L_n$. To the best of our knowledge, this concept has not been previously employed in any other reference dedicated to the examination of binary operators on $L_n$.

**Definition 3.15.** ([16]) Let $F: L_n^2 \to L_n$ be a binary operator. Given $z \in L$, the F-degree of $z$, denoted with $\deg_F(z)$, is defined as

$$ \deg_F(z) = \left| \{ (x, y) \in L_n^2 \setminus \{(z, z)\} \mid F(x, y) = F(z, z) \} \right|. \quad (40) $$

The degree sequence of $F$, denoted with $\deg_F$, is the increasing sequence of $\{\deg_F(z)\}_{z \in L_n}$.

They established an equivalence between specific properties of a binary operator that are simultaneously satisfied and idempotent discrete uninorms.

**Theorem 3.4.** ([14, Theorem 12], [16, Theorem 3.7]) For a binary operation $U: L_n^2 \to L_n$, the following statements are equivalent:

(i) $U$ is conservative, symmetric and increasing in each argument.

(ii) $U$ is conservative, increasing in each argument and $\deg_F = (0, 2, 4, \dots, 2n-2)$.

(iii) $U$ is an idempotent discrete uninorm.

With this result, they prove by induction the enumeration of the number of idempotent discrete uninorms. For both techniques, the enumeration is in the following result.

**Theorem 3.5.** The following statements hold:

(i) ([10, Theorem 4]) The number of idempotent uninorms on $L_n$, with $n \ge 2$, with neutral element $0 < e < n$, is given by

$$ |\mathcal{V}_n^{\text{IDE}(e)}| = \binom{n}{e}. $$

(ii) ([14, Theorem 14], [10, Theorem 4]) The total number of idempotent uninorms on $L_n$, with $n \ge 2$, is given by

$$ |\mathcal{V}_n^{\text{IDE}}| = \sum_{e=0}^{n} |\mathcal{V}_n^{\text{IDE}(e)}| = 2^n. $$

**Remark 3.6.** The formula for $|\mathcal{V}_n^{\text{IDE}(e)}|$ excludes the extreme values $e = 0$ and $e = n$, since it has been obtained with the characterization of Theorem 3.3, which also excludes them. Now, let us note that in the formula for $|\mathcal{V}_n^{\text{IDE}}|$ these extreme values are considered, since in order to define idempotent discrete uninorms they are not required to be proper discrete uninorms, and therefore it is possible to include the cases $e = 0$ and $e = n$.

Another class of discrete uninorms that are fully characterized are those whose underlying operators (the discrete t-norm $T_U$ and the discrete t-conorm $S_U$) are smooth. If $\mathcal{V}_n^{\text{STS}}$ denotes the set of these operators, Ruiz-Aguilera and Torrens determined in [11] its structure using the concept of Id-symmetric extension of a unary operator (see [10, Definition 4]). The general result is as follows.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

This result enabled the authors of [10] to determine the number of idempotent discrete uninorms with neutral element $e \in L_n \setminus \{0, n\}$ in particular, and also the number of idempotent discrete uninorms in general. Observe that in this result they convert the enumeration of these binary operators into a count of decreasing unary operators with fixed point $e$, whose enumeration is simpler. This enumeration can be found in Theorem 3.5.

The second approach is presented by Couceiro et al. in [14, 16]. In these papers, the authors reapply the methodology used in the study for the enumeration of several families of discrete aggregation functions described in Propositions 3.3 and 3.4, again from the perspective of the level curves of binary operators and their geometry. For that, they introduced a powerful tool, namely, the concept of the F-degree sequence of a binary operator $F: L_n^2 \to L_n$. To the best of our knowledge, this concept has not been previously employed in any other reference dedicated to the examination of binary operators on $L_n$.

**Definition 3.15.** ([16]) Let $F: L_n^2 \to L_n$ be a binary operator. Given $z \in L$, the F-degree of $z$, denoted with $\deg_F(z)$, is defined as

$$ \deg_F(z) = \left| \{ (x, y) \in L_n^2 \setminus \{(z, z)\} \mid F(x, y) = F(z, z) \} \right|. \quad (40) $$

The degree sequence of $F$, denoted with $\deg_F$, is the increasing sequence of $\{\deg_F(z)\}_{z \in L_n}$.

They established an equivalence between specific properties of a binary operator that are simultaneously satisfied and idempotent discrete uninorms.

**Theorem 3.4.** ([14, Theorem 12], [16, Theorem 3.7]) For a binary operation $U: L_n^2 \to L_n$, the following statements are equivalent:

(i) $U$ is conservative, symmetric and increasing in each argument.

(ii) $U$ is conservative, increasing in each argument and $\deg_F = (0, 2, 4, \dots, 2n - 2)$.

(iii) $U$ is an idempotent discrete uninorm.

With this result, they prove by induction the enumeration of the number of idempotent discrete uninorms. For both techniques, the enumeration is in the following result.

**Theorem 3.5.** The following statements hold:

(i) ([10, Theorem 4]) The number of idempotent uninorms on $L_n$, with $n \ge 2$, with neutral element $0 < e < n$, is given by

$$ |\mathcal{V}_n^{\text{IDE}(e)}| = \binom{n}{e}. $$

(ii) ([14, Theorem 14], [10, Theorem 4]) The total number of idempotent uninorms on $L_n$, with $n \ge 2$, is given by

$$ |\mathcal{V}_n^{\text{IDE}}| = \sum_{e=0}^{n} |\mathcal{V}_n^{\text{IDE}(e)}| = 2^n. $$

**Remark 3.6.** The formula for $|\mathcal{V}_n^{\text{IDE}(e)}|$ excludes the extreme values $e = 0$ and $e = n$, since it has been obtained with the characterization of Theorem 3.3, which also excludes them. Now, let us note that in the formula for $|\mathcal{V}_n^{\text{IDE}}|$ these extreme values are considered, since in order to define idempotent discrete uninorms they are not required to be proper discrete uninorms, and therefore it is possible to include the cases $e = 0$ and $e = n$.

Another class of discrete uninorms that are fully characterized are those whose underlying operators (the discrete t-norm $T_U$ and the discrete t-conorm $S_U$) are smooth. If $\mathcal{V}_n^{\text{STS}}$ denotes the set of these operators, Ruiz-Aguilera and Torrens determined in [11] its structure using the concept of Id-symmetric extension of a unary operator (see [10, Definition 4]). The general result is as follows.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

**Theorem 3.6.** ([11, Theorem 5]) Let $n \ge 2$. Let $U$ be a binary operator on $L_n$ with neutral element $0 < e < n$ and idempotent elements

$$ \mathrm{Ide}(U) = \{0 = a_0 < a_1 < \cdots < a_r = e = b_0 < b_1 < \cdots < b_s = n\}. $$

Then, the following statements are equivalent:

(i) $U$ is a uninorm on $L_n$ with smooth underlying operators $T_U$ and $S_U$.

(ii) There exists a decreasing function $g : [0, e] \to [e, n]$ with fixed point $e$ that satisfies simultaneously the following conditions:

(a) For all $a_i \in \mathrm{Ide}(U)$, there exists $b_j \in \mathrm{Ide}(U)$ such that $g(a_i) = b_j$.

(b) If $x \in [a_i, a_{i+1} - 1]$, then $g(x) = g(a_i)$, for all $i \in \{0, 1, \dots, r-1\}$.

(c) $U$ is given by

$$ U(x, y) = \begin{cases} T_U(x, y), & \text{if } (x, y) \in [0, e]^2, \\ S_U(x, y), & \text{if } (x, y) \in [e, n]^2, \\ \min\{x, y\}, & \text{if } (x, y) \in C_{e'} \text{, } y \le \bar{g}(x) \text{ and } x \le \bar{g}(0), \\ \max\{x, y\}, & \text{otherwise,} \end{cases} \quad (41) $$

where $\bar{g}$ is the only $\mathrm{Id}$-symmetric extension of $g$.

Once more, they characterized these operators starting from a decreasing unary operator, akin to their approach in Theorem 3.3 for idempotent discrete uninorms. Consequently, it suffices to carry out the counting of these decreasing unary operators, satisfying the conditions outlined in Theorem 3.6.

**Theorem 3.7.** ([11, Theorem 6]) The following statements hold.

(i) The number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ with neutral element $e \in [1, n-1]$, and $r+1$ idempotent elements on $[0, e]$ and $s+1$ idempotent elements on $[e, n]$ is given by

$$ \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (42) $$

(ii) The total number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ with neutral element $e \in [1, n-1]$, is given by

$$ \sum_{r=1}^{e} \sum_{s=1}^{n-1} \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (43) $$

(iii) The total number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ is given by

$$ |\mathcal{V}_n^{\mathrm{STS}}| = \sum_{e=1}^{n-1} \sum_{r=1}^{e} \sum_{s=1}^{n-1} \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (44) $$

**Remark 3.7.** In contrast to what has been discussed in Remark 3.6, to define discrete uninorms whose underlying operators are smooth, they are required to be proper discrete uninorms, since the extreme cases $e = 0$ and $e = n$ cause one of the two underlying operators to be ill-defined. For this reason, in the enumeration of statement (iii) of Theorem 3.7 the possibility of directly considering smooth discrete $t$-norms and smooth discrete $t$-conorms has been excluded.

Also with Theorem 3.6, they achieved an enumeration of discrete uninorms when the underlying operators are some specific examples of smooth discrete $t$-norms and discrete $t$-conorms.
**Theorem 3.6.** ([11, Theorem 5]) Let $n \ge 2$. Let $U$ be a binary operator on $L_n$ with neutral element $0 < e < n$ and idempotent elements

$$ \mathrm{Ide}(U) = \{0 = a_0 < a_1 < \cdots < a_r = e = b_0 < b_1 < \cdots < b_s = n\}. $$

Then, the following statements are equivalent:

(i) $U$ is a uninorm on $L_n$ with smooth underlying operators $T_U$ and $S_U$.

(ii) There exists a decreasing function $g : [0, e] \to [e, n]$ with fixed point $e$ that satisfies simultaneously the following conditions:

(a) For all $a_i \in \mathrm{Ide}(U)$, there exists $b_j \in \mathrm{Ide}(U)$ such that $g(a_i) = b_j$.

(b) If $x \in [a_i, a_{i+1} - 1]$, then $g(x) = g(a_i)$, for all $i \in \{0, 1, \dots, r-1\}$.

(c) $U$ is given by

$$ U(x, y) = \begin{cases} T_U(x, y), & \text{if } (x, y) \in [0, e]^2, \\ S_U(x, y), & \text{if } (x, y) \in [e, n]^2, \\ \min\{x, y\}, & \text{if } (x, y) \in C_{e'} \text{, } y \le \bar{g}(x) \text{ and } x \le \bar{g}(0), \\ \max\{x, y\}, & \text{otherwise,} \end{cases} \quad (41) $$

where $\bar{g}$ is the only $\mathrm{Id}$-symmetric extension of $g$.

Once more, they characterized these operators starting from a decreasing unary operator, akin to their approach in Theorem 3.3 for idempotent discrete uninorms. Consequently, it suffices to carry out the counting of these decreasing unary operators, satisfying the conditions outlined in Theorem 3.6.

**Theorem 3.7.** ([11, Theorem 6]) The following statements hold.

(i) The number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ with neutral element $e \in [1, n-1]$, and $r+1$ idempotent elements on $[0, e]$ and $s+1$ idempotent elements on $[e, n]$ is given by

$$ \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (42) $$

(ii) The total number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ with neutral element $e \in [1, n-1]$, is given by

$$ \sum_{r=1}^{e} \sum_{s=1}^{n-1} \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (43) $$

(iii) The total number of discrete uninorms $U = \langle T_U, e, S_U \rangle$ on $L_n$ with $n \ge 2$ having smooth underlying operators $T_U$ and $S_U$ is given by

$$ |\mathcal{V}_n^{\mathrm{STS}}| = \sum_{e=1}^{n-1} \sum_{r=1}^{e} \sum_{s=1}^{n-1} \binom{r+s}{r} \cdot \binom{e-1}{r-1} \cdot \binom{n-e-1}{s-1}. \quad (44) $$

**Remark 3.7.** In contrast to what has been discussed in Remark 3.6, to define discrete uninorms whose underlying operators are smooth, they are required to be proper discrete uninorms, since the extreme cases $e = 0$ and $e = n$ cause one of the two underlying operators to be ill-defined. For this reason, in the enumeration of statement (iii) of Theorem 3.7 the possibility of directly considering smooth discrete $t$-norms and smooth discrete $t$-conorms has been excluded.

Also with Theorem 3.6, they achieved an enumeration of discrete uninorms when the underlying operators are some specific examples of smooth discrete $t$-norms and discrete $t$-conorms.
**Proposition 3.19.** The following statements hold:

(i) ([11, Corollary 2]) The total number of discrete uninorms on $L_n$ such that their underlying operators are smooth and Archimedean is

$$|\mathcal{V}_n^{\text{STS}} \cap \mathcal{V}_n^{\text{ARC}}| = 2n - 2, \qquad (45)$$

for all $n \ge 2$.

(ii) ([11, Corollary 3]) The total number of discrete uninorms on $L_n$ such that their underlying operators are $T_U = T_M$ and $S_U = S_L$, is

$$\sum_{e=1}^{n-1} (e + 1) = \frac{n \cdot (n + 1)}{2} - 1, \qquad (46)$$

for all $n \ge 2$.

(iii) ([11, Corollary 4]) The total number of discrete uninorms on $L_n$, with $n \ge 1$, such that their underlying operators are $T_U = T_L$ and $S_U = S_M$, is

$$\sum_{e=1}^{n-1} (n - e + 1) = \frac{n \cdot (n + 1)}{2} - 1, \qquad (47)$$

for all $n \ge 2$.

The cases offered by this result allow to determine the number of discrete uninorms whose underlying operators are smooth and Archimedean, or smooth and idempotent, since $T_M$ (the minimum discrete t-norm) and $S_M$ (the maximum discrete t-conorm) are the only discrete t-norm and discrete t-conorm, respectively, that are smooth and idempotent, and $T_L$ and $S_L$ are the only discrete t-norm and discrete t-conorm, respectively, that are smooth and Archimedean. Ruiz-Aguilera and Torrens proposed in [11, Proposition 7] an erroneous formula for the enumeration of these operators, which we reformulate and prove.

**Proposition 3.20.** ([11, Proposition 7]) Let $n \ge 2$, and let us denote by $\mathcal{V}_n^{\text{STS}}$ and $\mathcal{V}_n^{\text{ARC,IDE}}$ the sets of discrete uninorms whose underlying operators are smooth and discrete uninorms whose underlying operators are Archimedean or idempotent, respectively, defined on $L_n$. Then,

$$|\mathcal{V}_n^{\text{STS}} \cap \mathcal{V}_n^{\text{ARC,IDE}}| = 2^n + n \cdot (n + 1) - 10, \qquad (48)$$

for all $n \ge 3$, with $|\mathcal{V}_2^{\text{STS}} \cap \mathcal{V}_2^{\text{ARC,IDE}}| = 2$ and $|\mathcal{V}_3^{\text{STS}} \cap \mathcal{V}_3^{\text{ARC,IDE}}| = 10$.

*Proof.* Let us focus when $n \ge 4$. Let $U$ be a discrete uninorm whose underlying operators are smooth and idempotent or smooth and Archimedean. In the following, we distinguish some cases according to the position of the neutral element $e$ of $U$.

* If $e = 1$, then the underlying discrete t-norm is defined on $L_1$, and on this domain we have the particularity that $T_L(x, y) = T_M(x, y)$ for all $x, y \in L_1$, so these two operators represent the same situation, and have $\{0, 1\}$ as idempotent elements. Then, we only need to focus on the behavior of $S_U$. We distinguish two cases:

- If $S_U$ is smooth and Archimedean, then $S_U$ has only 1 and $n$ as idempotent elements. Applying statement (i) of Theorem 3.7, this case is set up by taking $r = 1$, $s = 1$ and $e = 1$, which yields

$$\begin{pmatrix} 1+1 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} n-1-1 \\ 0 \end{pmatrix} = 2$$

operators.
**Proposition 3.19.** The following statements hold:

(i) ([11, Corollary 2]) The total number of discrete uninorms on $L_n$ such that their underlying operators are smooth and Archimedean is

$$|\mathcal{V}_n^{\text{STS}} \cap \mathcal{V}_n^{\text{ARC}}| = 2n - 2, \qquad (45)$$

for all $n \ge 2$.

(ii) ([11, Corollary 3]) The total number of discrete uninorms on $L_n$ such that their underlying operators are $T_U = T_M$ and $S_U = S_L$, is

$$\sum_{e=1}^{n-1} (e+1) = \frac{n \cdot (n+1)}{2} - 1, \qquad (46)$$

for all $n \ge 2$.

(iii) ([11, Corollary 4]) The total number of discrete uninorms on $L_n$, with $n \ge 1$, such that their underlying operators are $T_U = T_L$ and $S_U = S_M$, is

$$\sum_{e=1}^{n-1} (n - e + 1) = \frac{n \cdot (n + 1)}{2} - 1, \qquad (47)$$

for all $n \ge 2$.

The cases offered by this result allow to determine the number of discrete uninorms whose underlying operators are smooth and Archimedean, or smooth and idempotent, since $T_M$ (the minimum discrete t-norm) and $S_M$ (the maximum discrete t-conorm) are the only discrete t-norm and discrete t-conorm, respectively, that are smooth and idempotent, and $T_L$ and $S_L$ are the only discrete t-norm and discrete t-conorm, respectively, that are smooth and Archimedean. Ruiz-Aguilera and Torrens proposed in [11, Proposition 7] an erroneous formula for the enumeration of these operators, which we reformulate and prove.

**Proposition 3.20.** ([11, Proposition 7]) Let $n \ge 2$, and let us denote by $\mathcal{V}_n^{\text{STS}}$ and $\mathcal{V}_n^{\text{ARC,IDE}}$ the sets of discrete uninorms whose underlying operators are smooth and discrete uninorms whose underlying operators are Archimedean or idempotent, respectively, defined on $L_n$. Then,

$$|\mathcal{V}_n^{\text{STS}} \cap \mathcal{V}_n^{\text{ARC,IDE}}| = 2^n + n \cdot (n + 1) - 10, \qquad (48)$$

for all $n \ge 3$, with $|\mathcal{V}_2^{\text{STS}} \cap \mathcal{V}_2^{\text{ARC,IDE}}| = 2$ and $|\mathcal{V}_3^{\text{STS}} \cap \mathcal{V}_3^{\text{ARC,IDE}}| = 10$.

*Proof.* Let us focus when $n \ge 4$. Let $U$ be a discrete uninorm whose underlying operators are smooth and idempotent or smooth and Archimedean. In the following, we distinguish some cases according to the position of the neutral element $e$ of $U$.

* If $e = 1$, then the underlying discrete t-norm is defined on $L_1$, and on this domain we have the particularity that $T_L(x, y) = T_M(x, y)$ for all $x, y \in L_1$, so these two operators represent the same situation, and have $\{0, 1\}$ as idempotent elements. Then, we only need to focus on the behavior of $S_U$. We distinguish two cases:

- If $S_U$ is smooth and Archimedean, then $S_U$ has only 1 and $n$ as idempotent elements. Applying statement (i) of Theorem 3.7, this case is set up by taking $r = 1$, $s = 1$ and $e = 1$, which yields

$$\begin{pmatrix} 1+1 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} n-1-1 \\ 0 \end{pmatrix} = 2$$

operators.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

- If $S_U$ is smooth and idempotent, then it has $n - e + 1$ idempotent elements since it is defined on $[1, n]$. Therefore, applying statement (i) of Theorem 3.7, we have $r = 1, s = n - e$ and $e = 1$, giving

$$ \binom{1+n-e}{1} \cdot \binom{0}{0} \cdot \binom{n-1-1}{n-1-1} = n $$

operators.

* If $2 \le e \le n - 2$, then we distinguish four possible cases, bearing in mind that the above scenario, where different operators collapse and are the same, cannot occur.

- $T_U$ is smooth and Archimedean, and $S_U$ is smooth and Archimedean. Applying statement (i) of Theorem 3.7 with $r = 1$ and $s = 1$, we have

$$ \sum_{e=2}^{n-2} \binom{1+1}{1} \cdot \binom{e-1}{0} \cdot \binom{n-e-1}{0} = 2 \cdot (n-3) $$

operators on this case.

- $T_U$ is smooth and Archimedean, and $S_U$ is smooth and idempotent. Applying statement (i) of Theorem 3.7 with $r = 1$ and $s = n - e$, we have

$$ \sum_{e=2}^{n-2} \binom{n-e+1}{1} \cdot \binom{e-1}{0} \cdot \binom{n-e-1}{n-e-1} = \frac{n \cdot (n-1) - 6}{2} $$

operators on this case.

- $T_U$ is smooth and idempotent, and $S_U$ is smooth and Archimedean. Applying statement (i) of Theorem 3.7 with $r = e$ and $s = 1$, we have

$$ \sum_{e=2}^{n-2} \binom{e+1}{e} \cdot \binom{e-1}{e-1} \cdot \binom{n-e-1}{0} = \frac{n \cdot (n-1) - 6}{2} $$

operators on this case.

- $T_U$ is smooth and idempotent, and $S_U$ is smooth and idempotent. Applying statement (i) of Theorem 3.7 with $r = e$ and $s = n - e$, we have

$$ \sum_{e=2}^{n-2} \binom{e+n-e}{e} \cdot \binom{e-1}{e-1} \cdot \binom{n-e-1}{n-e-1} = 2^n - 2n - 2 $$

operators on this case.

* If $e = n - 1$. The case $e = n - 1$ is equivalent to the case $e = 1$, so $2 + n$ operators can be counted.

Gathering all the previous enumerations, which are mutually disjoint, the result holds. Finally, the cases $n \in \{2, 3\}$ can be obtained straightforwardly. $\square$

**Remark 3.8.** The formula proposed by Ruiz-Aguilera and Torrens in [11, Proposition 7] for the number of discrete uninorms whose underlying operators are smooth and Archimedean, or smooth and idempotent, is

$$ 2^n + n \cdot (n+3) - 2. $$

In their proof, they did not take into account that on $L_1$ it happens that $T_M(x, y) = T_L(x, y)$ and $S_M(x, y) = S_L(x, y)$ for all $x, y \in L_1$. Hence, when they made the final count, these cases are considered as different although they are in fact the same.

To summarize this section, we present in Table 6 the initial values of some of the sequences discussed throughout this section. Specifically, we have computationally determined the total count of discrete uninorms, $|V_n|$, for $n \le 9$. Using the formula from Remark 3.5, which relies on $T_n$ and $S_n$ (whose values are known for $n \le 13$), we have established $|V_n^{MIN}|$ and $|V_n^{MAX}|$ for $n \le 13$. Employing Theorem 3.5 and Theorem 3.7, we provide the values of $|V_n^{IDE}|$ and $|V_n^{STS}|$, respectively, for $n \le 13$. Lastly, we present some values of $|V_n^{STS} \cap V_n^{ARC,IDE}|$ with the corrected formula presented in Proposition 3.20.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

- If $S_U$ is smooth and idempotent, then it has $n - e + 1$ idempotent elements since it is defined on $[1, n]$. Therefore, applying statement (i) of Theorem 3.7, we have $r = 1, s = n - e$ and $e = 1$, giving

$$ \binom{1+n-e}{1} \cdot \binom{0}{0} \cdot \binom{n-1-1}{n-1-1} = n $$

operators.

* If $2 \le e \le n - 2$, then we distinguish four possible cases, bearing in mind that the above scenario, where different operators collapse and are the same, cannot occur.

- $T_U$ is smooth and Archimedean, and $S_U$ is smooth and Archimedean. Applying statement (i) of Theorem 3.7 with $r = 1$ and $s = 1$, we have

$$ \sum_{e=2}^{n-2} \binom{1+1}{1} \cdot \binom{e-1}{0} \cdot \binom{n-e-1}{0} = 2 \cdot (n-3) $$

operators on this case.

- $T_U$ is smooth and Archimedean, and $S_U$ is smooth and idempotent. Applying statement (i) of Theorem 3.7 with $r = 1$ and $s = n - e$, we have

$$ \sum_{e=2}^{n-2} \binom{n-e+1}{1} \cdot \binom{e-1}{0} \cdot \binom{n-e-1}{n-e-1} = \frac{n \cdot (n-1) - 6}{2} $$

operators on this case.

- $T_U$ is smooth and idempotent, and $S_U$ is smooth and Archimedean. Applying statement (i) of Theorem 3.7 with $r = e$ and $s = 1$, we have

$$ \sum_{e=2}^{n-2} \binom{e+1}{e} \cdot \binom{e-1}{e-1} \cdot \binom{n-e-1}{0} = \frac{n \cdot (n-1) - 6}{2} $$

operators on this case.

- $T_U$ is smooth and idempotent, and $S_U$ is smooth and idempotent. Applying statement (i) of Theorem 3.7 with $r = e$ and $s = n - e$, we have

$$ \sum_{e=2}^{n-2} \binom{e+n-e}{e} \cdot \binom{e-1}{e-1} \cdot \binom{n-e-1}{n-e-1} = 2^n - 2n - 2 $$

operators on this case.

* If $e = n - 1$. The case $e = n - 1$ is equivalent to the case $e = 1$, so $2 + n$ operators can be counted.

Gathering all the previous enumerations, which are mutually disjoint, the result holds. Finally, the cases $n \in \{2, 3\}$ can be obtained straightforwardly. □

**Remark 3.8.** The formula proposed by Ruiz-Aguilera and Torrens in [11, Proposition 7] for the number of discrete uninorms whose underlying operators are smooth and Archimedean, or smooth and idempotent, is

$$ 2^n + n \cdot (n + 3) - 2. $$

In their proof, they did not take into account that on $L_1$ it happens that $T_M(x, y) = T_L(x, y)$ and $S_M(x, y) = S_L(x, y)$ for all $x, y \in L_1$. Hence, when they made the final count, these cases are considered as different although they are in fact the same.

To summarize this section, we present in Table 6 the initial values of some of the sequences discussed throughout this section. Specifically, we have computationally determined the total count of discrete uninorms, $|V_n|$, for $n \le 9$. Using the formula from Remark 3.5, which relies on $T_n$ and $S_n$ (whose values are known for $n \le 13$), we have established $|V_n^{MIN}|$ and $|V_n^{MAX}|$ for $n \le 13$. Employing Theorem 3.5 and Theorem 3.7, we provide the values of $|V_n^{IDE}|$ and $|V_n^{STS}|$, respectively, for $n \le 13$. Lastly, we present some values of $|V_n^{STS} \cap V_n^{ARC,IDE}|$ with the corrected formula presented in Proposition 3.20.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 6**</caption><thead><tr><th>n</th><th>U<sub>n</sub></th><th>U<sub>n</sub><sup>MIN</sup></th><th>U<sub>n</sub><sup>MAX</sup></th><th>U<sub>n</sub><sup>IDE</sup></th><th>U<sub>n</sub><sup>STS</sup></th><th>U<sub>n</sub><sup>STS</sup> ∩ U<sub>n</sub><sup>ARC>IDE</sup></th></tr></thead><tbody><tr><td>1</td><td>2 = 2.10<sup>0</sup></td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>2</td><td>6 = 6.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td><td>—</td><td>4 = 4.10<sup>0</sup></td><td>2 = 2.10<sup>0</sup></td><td>2 = 6.10<sup>0</sup></td></tr><tr><td>3</td><td>22 = 2.210<sup>1</sup></td><td>4 = 4.10<sup>0</sup></td><td>—</td><td>8 = 8.10<sup>0</sup></td><td>10 = 1.10<sup>0</sup></td><td>10 = 1.10<sup>1</sup></td></tr><tr><td>4</td><td>92 = 9.210<sup>1</sup></td><td>16 = 1.610<sup>1</sup></td><td>—</td><td>16 = 1.610<sup>1</sup></td><td>38 = 3.810<sup>1</sup></td><td>26 = 2.610<sup>1</sup></td></tr><tr><td>5</td><td>426 = 4.2610<sup>2</sup></td><td>68 = 6.810<sup>1</sup></td><td>—</td><td>32 = 3.210<sup>1</sup></td><td>130 = 1.310<sup>2</sup></td><td>52 = 5.210<sup>1</sup></td></tr><tr><td>6</td><td>2,146 = 2.14610<sup>3</sup></td><td>312 = 3.1210<sup>2</sup></td><td>—</td><td>64 = 6.410<sup>1</sup></td><td>422 = 4.2210<sup>2</sup></td><td>96 = 9.610<sup>1</sup></td></tr><tr><td>7</td><td>11,634 = 1.163410<sup>4</sup></td><td>1,542 = 1.54210<sup>3</sup></td><td>—</td><td>128 = 1.2810<sup>2</sup></td><td>1,330 = 1.3310<sup>3</sup></td><td>174 = 1.7410<sup>2</sup></td></tr><tr><td>8</td><td>67,472 = 6.747210<sup>4</sup></td><td>8,188 = 8.18810<sup>3</sup></td><td>—</td><td>256 = 2.5610<sup>2</sup></td><td>4,118 = 4.11810<sup>3</sup></td><td>318 = 3.1810<sup>2</sup></td></tr><tr><td>9</td><td>417,650 = 4.176510<sup>5</sup></td><td>46,642 = 4.664210<sup>5</sup></td><td>—</td><td>512 = 5.1210<sup>2</sup></td><td>12,610 = 1.26110<sup>4</sup></td><td>592 = 5.9210<sup>2</sup></td></tr><tr><td>10</td><td>—</td><td>285,246 ≈ 2.852510<sup>5</sup></td><td>—</td><td>1,024 = 1.02410<sup>3</sup></td><td>38,342 = 3.834210<sup>4</sup></td><td>1,124 = 1.12410<sup>3</sup></td></tr><tr><td>11</td><td>—</td><td>1,881,718 ≈ 1.881710<sup>6</sup></td><td>—</td><td>2,048 = 2.04810<sup>3</sup></td><td>116,050 = 1.160510<sup>5</sup></td><td>2,170 = 2.1710<sup>3</sup></td></tr><tr><td>12</td><td>—</td><td>13,549,087 ≈ 1.354910<sup>7</sup></td><td>—</td><td>4,096 = 4.09610<sup>3</sup></td><td>≈ 3.50210<sup>5</sup></td><td>4,242 = 4.24210<sup>3</sup></td></tr><tr><td>13</td><td>—</td><td>109,153,102 ≈ 1.091510<sup>8</sup></td><td>—</td><td>8,192 = 8.19210<sup>3</sup></td><td>≈ 1.054710<sup>6</sup></td><td>8,364 = 8.36410<sup>3</sup></td></tr></tbody></table>

Enumeration of some classes of discrete uninorms for $n \le 13$. From left to right: discrete uninorms in general, obtained computationally; discrete uninorms such that $U(x, y) = \min\{x, y\}$ or $U(x, y) = \max\{x, y\}$, for all $(x, y) \in C_e$, given by Remark 3.5 which depends on the number of discrete t-norms in general, whose general formula is unknown; idempotent discrete uninorms, given by Theorem 3.5; discrete uninorms having smooth underlying operators, given by Theorem 3.7; and finally discrete uninorms having smooth and idempotent or smooth and Archimedean underlying operators, given by Proposition 3.20.

## 3.6. Discrete t-operators and nullnorms

A common practice when studying discrete operators is to generalize specific classes with the overarching objective of locating various operators within a more general class, and identifying shared properties among them. This practice is illustrated in the study of discrete t-operators and nullnorms, that were originally thought of as generalizations of discrete t-norms and discrete t-conorms, alternative to the generalization offered by discrete uninorms. We start with the definition of discrete t-operator.

**Definition 3.16.** ([7, Definition 3.2], [57, Definition 3.9]) A discrete t-operator is an associative and commutative discrete aggregation function $O: L_n^2 \to L_n$ such that the mappings $O_0(x) = O(0, x)$ and $O_n(x) = O(n, x)$, for all $x \in L_n$, are smooth.

Discrete t-operators were originally introduced in [7] by Mas et al., who proposed these operators including the smoothness in the definition. However, it is noteworthy that t-operators on $[0, 1]$ do not inherently include any hypothesis related to continuity in the whole unit square [58], since continuity is only imposed on the segments $\{0\} \times [0, 1]$ and $\{1\} \times [0, 1]$. Su and Liu in [57] observed the need to eliminate this assumption of smoothness from the definition and, consequently, referred to discrete t-operators as discrete pseudo-t-operators. Even so, for consistency with the existing nomenclature in the literature, we will refer by discrete t-operators in general to the class of discrete aggregation functions that satisfy the properties of Definition 3.16, and by smooth discrete t-operators to those discrete t-operators that are smooth on $L_n^2$.

The enumeration of discrete t-operators in the general case remains an open question. Nevertheless, it is worth highlighting that the characterization offered by Su and Liu as a combination of a discrete t-norm and a discrete t-conorm gives a path to derive a formula for their enumeration.

**Theorem 3.8.** ([57, Corollary 3.10]) A binary operator $O: L_n^2 \to L_n$ is a discrete t-operator if, and only if, there exist a discrete t-conorm $S$ on $[0, k]$ and a discrete t-norm $T$ on $[k, n]$, where $k = O(0, n)$, such that $O$ is given by

$$O(x, y) = \begin{cases} S(x, y), & \text{if } (x, y) \in [0, k]^2, \\ T(x, y), & \text{if } (x, y) \in [k, n]^2, \\ k, & \text{otherwise,} \end{cases} \quad (49)$$

for all $x, y \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

<table><caption>**Table 6**</caption><thead><tr><th>n</th><th>U<sub>n</sub><sup>MIN</sup></th><th>U<sub>n</sub><sup>MAX</sup></th><th>U<sub>n</sub><sup>IDE</sup></th><th>U<sub>n</sub><sup>STS</sup></th><th>U<sub>n</sub><sup>STS</sup> ∩ U<sub>n</sub><sup>ARC,IDE</sup></th></tr></thead><tbody><tr><td>1</td><td>2 = 2.10<sup>0</sup></td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>2</td><td>6 = 6.10<sup>0</sup></td><td>1 = 1.10<sup>0</sup></td><td>4 = 4.10<sup>0</sup></td><td>2 = 2.10<sup>0</sup></td><td>2 = 6.10<sup>0</sup></td></tr><tr><td>3</td><td>22 = 2.210<sup>1</sup></td><td>4 = 4.10<sup>0</sup></td><td>8 = 8.10<sup>0</sup></td><td>10 = 1.10<sup>0</sup></td><td>10 = 1.10<sup>1</sup></td></tr><tr><td>4</td><td>92 = 9.210<sup>1</sup></td><td>16 = 1.610<sup>1</sup></td><td>16 = 1.610<sup>1</sup></td><td>38 = 3.810<sup>1</sup></td><td>26 = 2.610<sup>1</sup></td></tr><tr><td>5</td><td>426 = 4.2610<sup>2</sup></td><td>68 = 6.810<sup>1</sup></td><td>32 = 3.210<sup>1</sup></td><td>130 = 1.310<sup>2</sup></td><td>52 = 5.210<sup>1</sup></td></tr><tr><td>6</td><td>2,146 = 2.14610<sup>3</sup></td><td>312 = 3.1210<sup>2</sup></td><td>64 = 6.410<sup>1</sup></td><td>422 = 4.2210<sup>2</sup></td><td>96 = 9.610<sup>1</sup></td></tr><tr><td>7</td><td>11,634 = 1.163410<sup>4</sup></td><td>1,542 = 1.54210<sup>3</sup></td><td>128 = 1.2810<sup>2</sup></td><td>1,330 = 1.3310<sup>3</sup></td><td>174 = 1.7410<sup>2</sup></td></tr><tr><td>8</td><td>67,472 = 6.747210<sup>4</sup></td><td>8,188 = 8.18810<sup>3</sup></td><td>256 = 2.5610<sup>2</sup></td><td>4,118 = 4.11810<sup>3</sup></td><td>318 = 3.1810<sup>2</sup></td></tr><tr><td>9</td><td>417,650 = 4.176510<sup>5</sup></td><td>46,642 = 4.664210<sup>5</sup></td><td>512 = 5.1210<sup>2</sup></td><td>12,610 = 1.26110<sup>4</sup></td><td>592 = 5.9210<sup>2</sup></td></tr><tr><td>10</td><td>—</td><td>285,246 ≈ 2.852510<sup>5</sup></td><td>1,024 = 1.02410<sup>3</sup></td><td>38,342 = 3.834210<sup>4</sup></td><td>1,124 = 1.12410<sup>3</sup></td></tr><tr><td>11</td><td>—</td><td>1,881,718 ≈ 1.881710<sup>6</sup></td><td>2,048 = 2.04810<sup>3</sup></td><td>116,050 = 1.160510<sup>5</sup></td><td>2,170 = 2.1710<sup>3</sup></td></tr><tr><td>12</td><td>—</td><td>13,549,087 ≈ 1.354910<sup>7</sup></td><td>4,096 = 4.09610<sup>3</sup></td><td>≈ 3.50210<sup>5</sup></td><td>4,242 = 4.24210<sup>3</sup></td></tr><tr><td>13</td><td>—</td><td>109,153,102 ≈ 1.091510<sup>8</sup></td><td>8,192 = 8.19210<sup>3</sup></td><td>≈ 1.054710<sup>6</sup></td><td>8,364 = 8.36410<sup>3</sup></td></tr></tbody></table>

Enumeration of some classes of discrete uninorms for $n \le 13$. From left to right: discrete uninorms in general, obtained computationally; discrete uninorms such that $U(x, y) = \min\{x, y\}$ or $U(x, y) = \max\{x, y\}$, for all $(x, y) \in C_e$, given by Remark 3.5 which depends on the number of discrete t-norms in general, whose general formula is unknown; idempotent discrete uninorms, given by Theorem 3.5; discrete uninorms having smooth underlying operators, given by Theorem 3.7; and finally discrete uninorms having smooth and idempotent or smooth and Archimedean underlying operators, given by Proposition 3.20.

## 3.6. Discrete t-operators and nullnorms

A common practice when studying discrete operators is to generalize specific classes with the overarching objective of locating various operators within a more general class, and identifying shared properties among them. This practice is illustrated in the study of discrete t-operators and nullnorms, that were originally thought of as generalizations of discrete t-norms and discrete t-conorms, alternative to the generalization offered by discrete uninorms. We start with the definition of discrete t-operator.

**Definition 3.16.** ([7, Definition 3.2], [57, Definition 3.9]) A discrete t-operator is an associative and commutative discrete aggregation function $O: L_n^2 \to L_n$ such that the mappings $O_0(x) = O(0, x)$ and $O_n(x) = O(n, x)$, for all $x \in L_n$, are smooth.

Discrete t-operators were originally introduced in [7] by Mas et al., who proposed these operators including the smoothness in the definition. However, it is noteworthy that t-operators on $[0, 1]$ do not inherently include any hypothesis related to continuity in the whole unit square [58], since continuity is only imposed on the segments $\{0\} \times [0, 1]$ and $\{1\} \times [0, 1]$. Su and Liu in [57] observed the need to eliminate this assumption of smoothness from the definition and, consequently, referred to discrete t-operators as discrete pseudo-t-operators. Even so, for consistency with the existing nomenclature in the literature, we will refer by discrete t-operators in general to the class of discrete aggregation functions that satisfy the properties of Definition 3.16, and by smooth discrete t-operators to those discrete t-operators that are smooth on $L_n^2$.

The enumeration of discrete t-operators in the general case remains an open question. Nevertheless, it is worth highlighting that the characterization offered by Su and Liu as a combination of a discrete t-norm and a discrete t-conorm gives a path to derive a formula for their enumeration.

**Theorem 3.8.** ([57, Corollary 3.10]) A binary operator $O: L_n^2 \to L_n$ is a discrete t-operator if, and only if, there exist a discrete t-conorm $S$ on $[0, k]$ and a discrete t-norm $T$ on $[k, n]$, where $k = O(0, n)$, such that $O$ is given by

$$O(x, y) = \begin{cases} S(x, y), & \text{if } (x, y) \in [0, k]^2, \\ T(x, y), & \text{if } (x, y) \in [k, n]^2, \\ k, & \text{otherwise,} \end{cases} \quad (49)$$

for all $x, y \in L_n$.
A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

Let $\mathcal{O}_n$ be the set of all discrete t-operators on $L_n$. A formula for $|\mathcal{O}_n|$ can be obtained in a manner similar to the formula derived for some classes of discrete uninorms, explained in Remark 3.5, also depending on the number of discrete t-norms and discrete t-conorms. This formula is given by

$$ |\mathcal{O}_n| = |S_n| + |\mathcal{T}_n| + \sum_{k=1}^{n-1} |S_k| \cdot |\mathcal{T}_{n-k}| = 2 \cdot |\mathcal{T}_n| + \sum_{e=1}^{n-1} |\mathcal{T}_e| \cdot |\mathcal{T}_{n-e}|, \quad (50) $$

which interestingly coincides with the formula for $|\mathcal{U}_n^{\text{MIN}}|$ and $|\mathcal{U}_n^{\text{MAX}}|$. But further consequences can be drawn from Theorem 3.8. Mainly, that a discrete t-operator is smooth if, and only if, the underlying discrete t-norm and discrete t-conorm are smooth, a result also reached by Mas et al. in [7] with discrete t-operators. If we substitute $|\mathcal{T}_n|$ in Equation (50) with $|\mathcal{T}_n^{\text{SMT}}|$, we obtain a formula for the number of smooth discrete t-operators, which is equivalent to the number of discrete t-operators. And finally, if we look for the number of discrete t-operators that are idempotent, given that there is only one idempotent discrete t-norm and t-conorm, we can obtain a formula for this class of discrete t-operators.

**Proposition 3.21.** ([7, Corollary 3.8]) For $n \ge 2$, the following statements hold:

(i) If $\mathcal{O}_n^{\text{SMT}}$ denotes the set of smooth discrete t-operators, then $|\mathcal{O}_n^{\text{SMT}}| = 2^n + (n-1) \cdot 2^{n-2}$.

(ii) If $\mathcal{O}_n^{\text{IDE}}$ denotes the set of idempotent discrete t-operators, then $|\mathcal{O}_n^{\text{IDE}}| = n+1$.

We finish this section with discrete nullnorms, introduced and characterized by Riera and Torrens in [59].

**Definition 3.17.** ([59, Definition 2.4]) A discrete nullnorm is an associative and commutative discrete aggregation function $H: L_n^2 \to L_n$ which has an absorbing element $k \in L_n$ satisfying $H(0, x) = x$, for all $x \le k$, and $H(n, x) = x$, for all $x \ge k$.

In a similar way to what happens on [0, 1], discrete nullnorms and discrete t-operators have different definitions (albeit with shared properties), Riera and Torrens characterized in [59, Proposition 2.7] discrete nullnorms with the same result as the one stated in Theorem 3.8; that is, an operator $H$ is a discrete nullnorm if, and only if, it satisfies Equation (49). Given that these two classes of operators are essentially the same, it is evident that the enumerations discussed previously are also applicable to discrete nullnorms. Specifically, the number of discrete nullnorms, $|H_n|$, is governed by the formula of Equation (50). Moreover, $|H_n^{\text{SMT}}|$ and $|H_n^{\text{IDE}}|$, representing the number of smooth and idempotent discrete nullnorms, respectively, are determined in Proposition 3.21. Table 7 provides examples of the enumerations discussed in this section for $n \le 13$.

# 4. Enumeration of discrete implications and some subclasses

Discrete implications extend the concept of implication from Boolean logic, mirroring how discrete negations, discrete t-norms and discrete t-conorms broaden the notions of negation, conjunction and disjunction in Boolean logic, respectively. The two surveys [60, 61] compile a large number of properties of fuzzy implication functions<fnref n="2" />, while in the two surveys [62, 63] we find sections dedicated to discrete implications. Initially, the concept of discrete implication was proposed by Nachtegael and Kerre [6] under the name of *implicator*, and was studied in detail by Mas et al. in [64] establishing the usual nomenclature.

**Definition 4.1.** ([64, Definition 3]) A discrete implication is a binary operation $I: L_n^2 \to L_n$ such that it is decreasing in the first argument, increasing in the second argument, $I(0, 0) = I(n, n) = n$ and $I(n, 0) = 0$.

The cardinality of these operators already attracted the attention of Nachtegael and Kerre in their seminal paper [6]. Computationally, they determined the number of discrete implications on $L_n$ for $n \le 5$, and it was not until years later that Munar et al. in [19] determined a general formula for the enumeration of discrete implications. Their procedure was based on establishing a bijection between discrete implications and discrete conjunctions.

<footnote n="2">²When the domain is [0, 1], the implications defined on it are generally referred to as *fuzzy implications*.</footnote>