# Chebyshev distances associated to the second members of Systems of Max-product/Lukasiewicz Fuzzy Relational Equations

Ismaël Baaj

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Ismaël Baaj. Chebyshev distances associated to the second members of Systems of Max-product/Lukasiewicz Fuzzy Relational Equations. 2023. hal-03962307

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Distributed under a Creative Commons Attribution 4.0 International License
CHEBYSHEV DISTANCES ASSOCIATED TO THE SECOND
MEMBERS OF SYSTEMS OF MAX-PRODUCT/LUKASIEWICZ
FUZZY RELATIONAL EQUATIONS

A PREPRINT

Ismaël Baaj

Univ. Artois, CNRS, CRIL, F-62300 Lens, France

baaj@cril.fr

ABSTRACT

In this article, we study the inconsistency of a system of max-product fuzzy relational equations and of a system of max-Lukasiewicz fuzzy relational equations. For a system of max – min fuzzy relational equations $A\square_{\min}^{\max} x = b$ and using the $L_{\infty}$ norm, (Baaj, 2023) showed that the Chebyshev distance $\Delta = \inf_{c \in C} \|b-c\|$, where $C$ is the set of second members of consistent systems defined with the same matrix $A$, can be computed by an explicit analytical formula according to the components of the matrix $A$ and its second member $b$. In this article, we give analytical formulas analogous to that of (Baaj, 2023) to compute the Chebyshev distance associated to the second member of a system of max-product fuzzy relational equations and that associated to the second member of a system of max-Lukasiewicz fuzzy relational equations.

**Keywords** Fuzzy set theory ; Systems of fuzzy relational equations ; Chebyshev approximations

# 1 Introduction

Artificial Intelligence (AI) applications based on systems of fuzzy relational equations emerged thanks to (Sanchez, 1976, 1977)'s seminal work on solving systems of max – min fuzzy relational equations. (Sanchez, 1976) gave necessary and sufficient conditions for a system to be consistent i.e., to have solutions. (Sanchez, 1977) showed that if the system is consistent, it has a greater solution and many minimal solutions, and he then described the complete set of solutions of the system.

However, the inconsistency of these systems remains difficult to address. While the majority of the approaches have investigated how to determine approximate solutions of an inconsistent system (Di Nola et al., 1989; Gottwald, 1986; Klir and Yuan, 1994; Pedrycz, 1983, 1990a; Wangming, 1986; Wen et al., 2022; Wu et al., 2021, 2022; Xiao et al., 2019), Pedrycz highlighted another strategy in (Pedrycz, 1990b). Given an inconsistent system, Pedrycz proposed to slightly modify its second member to obtain a consistent system. Some authors proposed algorithms for his procedure (Cuninghame-Green and Cechlárová, 1995; Li and Fang, 2010). In a recent preprint (Baaj, 2023), for an inconsistent system of max – min fuzzy relational equations $A\square_{\min}^{\max} x = b$, the author gave an explicit analytical formula to compute, using the $L_{\infty}$ norm, the Chebyshev distance $\Delta = \inf_{c \in C} \|b-c\|$ where $C$ is the set of second members of consistent systems defined with the same matrix $A$. As a Chebyshev approximation of the second member $b$ is a vector $c$ such that $\|b-c\| = \Delta$ and the system $A\square_{\min}^{\max} x = c$ is a consistent system (Cuninghame-Green and Cechlárová, 1995; Li and Fang, 2010), the formula of $\Delta$ led the author of (Baaj, 2023) to give the complete description of the structure of the set of Chebyshev approximations of $b$. He then described the approximate solutions set of the considered inconsistent system, which is the set of solutions of consistent systems $A\square_{\min}^{\max} x = c$, where $c$ is a Chebyshev approximation of $b$.

In our opinion, it would be relevant to study how these results can be extended to systems of max-T fuzzy relational equations, where T is a continuous t-norm. As it is well-known, many applications are based on systems of fuzzy relational equations with the max-product composition or the max-Lukasiewicz composition e.g., (Di Nola et al., 1989; Hirota and Pedrycz, 1999; Nobuhara et al., 2006).
In this article, we tackle the problem of the inconsistency of a system of max-product fuzzy relational equations and of a system of max-Lukasiewicz fuzzy relational equations. We extend to systems of max-$T$ fuzzy relational equations, where $T$ is a continuous t-norm, the definition and the properties of the idempotent application $F$, see (11), introduced in (Baaj, 2023). This application was initially proposed as a reformulation of (Sanchez, 1976)'s result for checking if a system of max-min fuzzy relational equations defined with a fixed matrix and a given vector used as second member is a consistent system. The idempotence and the right-continuity of the application $F$ lets us extend a fundamental result given for max-min composition (Theorem 1 of (Cuninghame-Green and Cechlárová, 1995)) to max-$T$ composition (Theorem 1), and to obtain the greatest Chebyshev approximation of the second member of such a system (Corollary 1).

Then, we give an explicit analytical formula to compute the Chebyshev distance associated to the second member of a system of max-product fuzzy relational equations (Theorem 4). In the case of a system of max-Lukasiewicz fuzzy relational equations, an analogous analytical formula is also given (Theorem 5). Each Chebyshev distance is computed according to the components of the matrix and the second member of the considered system.

The article is structured as follows. In (Section 2), we begin by reminding the necessary background on solving a system of max-$T$ fuzzy relational equations where $T$ is a continuous t-norm. We extend the idempotent application $F$ of (Baaj, 2023) to a system of max-$T$ fuzzy relational equations. We remind the definition of the Chebyshev distance associated to the second member of a system of max-$T$ fuzzy relational equations and extend to such a system a fundamental result given for max-min composition (Theorem 1 of (Cuninghame-Green and Cechlárová, 1995)). In (Section 3), we study the solving of two inequalities, which are involved in the computation of the Chebyshev distance associated to the second member of a system of max-product fuzzy relational equations and that associated to the second member of a system of max-Lukasiewicz fuzzy relational equations. In (Section 4), we give the explicit analytical formulas to compute the two Chebyshev distances. Finally, we conclude with some perspectives.

Our results are illustrated with examples (note that in all examples, the numbers are rounded to two decimal places).

## 2 Background

Throughout this section, $T$ denotes a continuous t-norm.

In what follows, we begin by reminding the definition and properties of a continuous t-norm and its associated residual implicator. To the t-norm defined by the usual product, we make explicit its associated residual implicator, as well as the one associated to the t-norm of Lukasiewicz.

We then extend to systems of max-$T$ fuzzy relational equations the idempotent application $F$ introduced in (Baaj, 2023) for checking if a system of max-min fuzzy relational equations defined with a fixed matrix and a given vector used as second member is a consistent system. Finally, we remind the definition of the Chebyshev distance associated to the second member of a system of max-$T$ fuzzy relational equations and extend to such a system a fundamental result proven in (Cuninghame-Green and Cechlárová, 1995) for max-min composition.

### 2.1 Notations

The following notations were given in (Baaj, 2023) for the case of a system of max-min fuzzy relational equations and we reuse them in this article.

By $[0, 1]^{n \times m}$ we denote the set of matrices of size $(n, m)$ i.e., $n$ rows and $m$ columns, whose components are in $[0, 1]$. In particular:

* $[0, 1]^{n \times 1}$ denotes the set of column vectors of $n$ components,

* $[0, 1]^{1 \times m}$ denotes the set of row matrices of $m$ components.

On the set $[0, 1]^{n \times m}$, we use the order relation $\le$ defined by:

$$A \le B \quad \text{iff we have} \quad a_{ij} \le b_{ij} \quad \text{for all} \quad 1 \le i \le n, 1 \le j \le m,$$

where $A = [a_{ij}]_{1 \le i \le n, 1 \le j \le m}$ and $B = [b_{ij}]_{1 \le i \le n, 1 \le j \le m}$.

For $x, y, z, u, \delta \in [0, 1]$, we put:

* $x^+ = \max(x, 0)$,

* $\bar{z}(\delta) = \min(z + \delta, 1)$,

* $z(\delta) = \max(z - \delta, 0) = (z - \delta)^+$.
and we have the following equivalence in $[0, 1]$:

$$ |x - y| \le \delta \iff \underline{x}(\delta) \le y \le \bar{x}(\delta). \qquad (1) $$

For our work, to a column-vector $b = [b_i]_{1 \le i \le n}$ and a number $\delta \in [0, 1]$, we associate two column-vectors:

$$ \underline{b}(\delta) = [(b_i - \delta)^+]_{1 \le i \le n} \quad \text{and} \quad \bar{b}(\delta) = [\min(b_i + \delta, 1)]_{1 \le i \le n}. \qquad (2) $$

These vectors $\underline{b}(\delta)$ and $\bar{b}(\delta)$ were already introduced e.g., in (Cuninghame-Green and Cechlárová, 1995) (with others notations) and in (Li and Fang, 2010).

Then, from (1), we deduce for any $c = [c_i]_{1 \le i \le n} \in [0, 1]^{n \times 1}$:

$$ \|b - c\| \le \delta \iff \underline{b}(\delta) \le c \le \bar{b}(\delta). \qquad (3) $$

where $\|b - c\| = \max_{1 \le i \le n} |b_i - c_i|$.

## 2.2 T-norms and their associated residual implicators

A triangular-norm (t-norm, see (Klement et al., 2013)) is a map $T: [0, 1] \times [0, 1] \to [0, 1]$, which satisfies:

* $T$ is commutative: $T(x, y) = T(y, x)$,

* $T$ is associative: $T(x, T(y, z)) = T(T(x, y), z)$,

* $T$ is increasing with respect to the second variable: $y \le z \Rightarrow T(x, y) \le T(x, z)$,

* $T$ has 1 as neutral element: $T(x, 1) = x$.

The residual implicator associated to the t-norm $T$ is the map $\mathcal{I}_T: [0, 1] \times [0, 1] \to [0, 1]: (x, y) \mapsto \mathcal{I}_T(x, y) = \sup\{z \in [0, 1] \mid T(x, z) \le y\}$.

The main properties of the residual implicator $\mathcal{I}_T$ of a continuous t-norm $T$ are: for all $a, b \in [0, 1]$, we have:

* $\mathcal{I}_T(a, b) = \max\{z \in [0, 1] \mid T(a, z) \le b\}$. Therefore, $T(a, \mathcal{I}_T(a, b)) \le b$.

* $\mathcal{I}_T$ is left-continuous and decreasing in its first argument as well as right-continuous and increasing in its second argument.

* For all $z \in [0, 1]$, we have:

$$ T(a, z) \le b \iff z \le \mathcal{I}_T(a, b). $$

* We have $b \le \mathcal{I}_T(a, T(a, b))$.

The t-norm defined by the usual product is denoted by $T_P$. Its associated residual implicator is the Goguen product, and we have:

$$ T_P(x, y) = x \cdot y \quad ; \quad \mathcal{I}_{T_P}(x, y) = x \underset{GG}{\longrightarrow} y = \begin{cases} 1 & \text{if } x \le y \\ \frac{y}{x} & \text{if } x > y \end{cases} \qquad (4) $$

Lukasiewicz's t-norm is denoted by $T_L$ and we have:

$$ T_L(x, y) = \max(x + y - 1, 0) = (x + y - 1)^+ \quad ; \quad \mathcal{I}_{T_L}(x, y) = x \underset{L}{\longrightarrow} y = \min(1 - x + y, 1). \qquad (5) $$

In the rest of this section, we set a continuous t-norm denoted $T$.

## 2.3 Solving systems of max-$T$ fuzzy relational equations

Let $A = [a_{ij}] \in [0, 1]^{n \times m}$ be a matrix of size $(n, m)$ and $b = [b_i] \in [0, 1]^{n \times 1}$ be a vector of $n$ components. The system of max-$T$ fuzzy relational equations associated to $(A, b)$ is of the form:

$$ (S) : A \Box_T^{\max} x = b, \qquad (6) $$

where $x = [x_j]_{1 \le j \le m} \in [0, 1]^{m \times 1}$ is an unknown vector of $m$ components and the operator $\Box_T^{\max}$ is the matrix product that uses the continuous t-norm $T$ as the product and max as the addition. Equivalently, the system can also be written as:

$$ \max_{1 \le j \le m} T(a_{ij}, x_j) = b_i, \forall i \in \{1, 2, \dots, n\}. $$
The studies on systems of fuzzy relational equations adopt two equivalent notation conventions, which differ in their representation of the unknown part and the second member: either as column vectors or as row vectors. The transpose map allows us to switch from one notation to the other.

To check if the system (S), see (6), is consistent, we compute the following vector:

$$e = A^t \Box_{\mathcal{I}_T}^{\min} b, \qquad (7)$$

where $A^t$ is the transpose of A and the matrix product $\Box_{\mathcal{I}_T}^{\min}$ uses the residual implicator $\mathcal{I}_T$ (associated to T) as the product and min as the addition. The vector e is the potential maximal solution of the system (S).

Thanks to (Sanchez, 1976)'s seminal work on the solving of systems max – min fuzzy relational equations, and (Pedrycz, 1982, 1985) and (Miyakoshi and Shimbo, 1985) (using other t-norms than min), we have the following equivalence:

$$A \Box_{\mathcal{I}_T}^{\max} x = b \text{ is consistent} \iff A \Box_{\mathcal{I}_T}^{\max} e = b. \qquad (8)$$

In the rest of this subsection, we will give (Proposition 1) and show that the proof of its second statement implies the equivalence (8).

The set of solutions of the system (S) is denoted by:

$$S = S(A, b) = \{v \in [0, 1]^{m \times 1} \mid A \Box_{\mathcal{I}_T}^{\max} v = b\}. \qquad (9)$$

The structure of the solution set was described by (Sanchez, 1977) (with the max – min composition) and (Di Nola et al., 1982, 1984, 1989).

**Example 1.** Let:

$$A = \begin{bmatrix} 0.31 & 0.49 & 0.76 \\ 0.34 & 0.9 & 0.15 \\ 0.94 & 0.47 & 0.05 \end{bmatrix} \quad \text{and} \quad b = \begin{bmatrix} 0.73 \\ 0.84 \\ 0.61 \end{bmatrix}. \qquad (10)$$

To check if the systems $A\Box_{\mathcal{I}_T}^{\max} x = b$ and $A\Box_{\mathcal{I}_L}^{\max} x = b$ are consistent, we compute their respective potential greatest solution:

$$e_{T_P} = A^t \Box_{\mathcal{I}_{T_P}}^{\min} b = \begin{bmatrix} 0.65 \\ 0.93 \\ 0.96 \end{bmatrix} \quad \text{and} \quad e_{T_L} = A^t \Box_{\mathcal{I}_{T_L}}^{\min} b = \begin{bmatrix} 0.67 \\ 0.94 \\ 0.97 \end{bmatrix}.$$

These two systems are consistent because:

$$A \Box_{\mathcal{I}_P}^{\max} e_{T_P} = A \Box_{\mathcal{I}_L}^{\max} e_{T_L} = b.$$

In (Baaj, 2023), the author introduced an idempotent application denoted F, to check if a system of max – min fuzzy relational equations defined with a fixed matrix $A = [a_{ij}]_{1 \le i \le n, 1 \le j \le m} \in [0, 1]^{n \times m}$ and a given vector $c \in [0, 1]^{n \times 1}$ used as second member is a consistent system. The application is a reformulation of (Sanchez, 1976)'s result. For systems of max – T fuzzy relational equations which uses a continuous t-norm T, we extend the definition of the application F as follows:

$$F: [0, 1]^{n \times 1} \to [0, 1]^{n \times 1} : c = [c_i] \mapsto F(c) = [F(c)_i] \qquad (11)$$

where:

$$\forall i \in \{1, 2, \dots, n\}, F(c)_i = \max_{1 \le j \le m} T(a_{ij}, \min_{1 \le k \le n} \mathcal{I}_T(a_{kj}, c_k)). \qquad (12)$$

The properties of the application F given in (Baaj, 2023) for a system of max – min fuzzy relational equations are valid for systems of max – T fuzzy relational equations:

**Proposition 1.**

1. $\forall c \in [0, 1]^{n \times 1}, F(c) \le c$.

2. For any vector $c \in [0, 1]^{n \times 1}$ we have:

$$F(c) = c \iff \text{the system } A \Box_{\mathcal{I}_T}^{\max} x = c \text{ is consistent.}$$

3. F is idempotent i.e., $\forall c \in [0, 1]^{n \times 1}, F(F(c)) = F(c)$.

4. F is increasing and right-continuous.

The application F being right-continuous at a point $c \in [0, 1]^{n \times 1}$ means: for any sequence $(c^{(k)})$ in $[0, 1]^{n \times 1}$ such that $(c^{(k)})$ converges to $c$ when $k \to \infty$ and verifying $\forall k, c^{(k)} \ge c$, we have:
$$F(c^{(k)}) \to F(c) \text{ when } k \to \infty.$$

*Proof.* We adapt the proofs given in (Baaj, 2023) for the max-min composition to the max-T composition in the following way:

The first assertion is deduced from the inequality $T(a, \mathcal{I}_T(a, b)) \le b$.

For the second statement, if we suppose that $F(c) = c$, it means that $e = A^t \Box_{\mathcal{I}_T}^{\min} c$ is a solution of the system $A \Box_{\mathcal{I}_T}^{\max} x = c$ (so it is consistent).

For the proof of the implication $\Leftarrow$, we use the general inequality:

$$\forall x \in [0, 1]^{m \times 1}, x \le A^t \Box_{\mathcal{I}_T}^{\min} (A \Box_{\mathcal{I}_T}^{\max} x),$$

which is deduced from the inequality $b \le \mathcal{I}_T(a, T(a, b))$. We also obtain that $e = A^t \Box_{\mathcal{I}_T}^{\min} c$ is the greatest solution of the system $A \Box_{\mathcal{I}_T}^{\max} x = c$. Clearly, we have also proven the equivalence (8).

Thanks to the two assertions above, the idempotence of the application $F$ is proved as in the case of the max-min composition.

It is easy to prove that the application $F$ is increasing. The right-continuity of the residual implicator $\mathcal{I}_T$ in its second argument implies the right-continuity of the application $F$.

<figure><img src="image_3.png" /></figure>

**Example 2.** (continued) Let $c = \begin{bmatrix} 0.74 \\ 0.66 \\ 0.62 \end{bmatrix}$. When we use any of the two t-norms $T_P$ or $T_L$, we have the equality

$F(c) = c$. So the systems $A \Box_{T_P}^{\max} x = c$ and $A \Box_{T_L}^{\max} x = c$ are consistent.

## 2.4 Chebyshev distance associated to the second member of a system of max-T fuzzy relational equations

To the matrix $A$ and the second member $b$ of the system (S) of max-T fuzzy relational equations, see (6), let us associate the set of vectors $c = [c_i] \in [0, 1]^{n \times 1}$ such that the system $A \Box_{\mathcal{I}_T}^{\max} x = c$ is consistent:

$$\mathcal{C} = \{c = [c_i] \in [0, 1]^{n \times 1} \mid A \Box_{\mathcal{I}_T}^{\max} x = c \text{ is consistent}\}. \quad (13)$$

This set allows us to define the Chebyshev distance associated to the second member $b$ of the system (S).

**Definition 1.** The Chebyshev distance associated to the second member $b$ of the system (S) : $A \Box_{\mathcal{I}_T}^{\max} x = b$ is:

$$\Delta = \Delta(A, b) = \inf_{c \in \mathcal{C}} \|b - c\| \quad (14)$$

where:

$$\|b - c\| = \max_{1 \le i \le n} |b_i - c_i|.$$

We have the following fundamental result, proven for max-min composition in (Cuninghame-Green and Cechlárová, 1995), that we extend for max-T composition:

**Theorem 1.**

$$\Delta = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}. \quad (15)$$

The proof of (Theorem 1) is given in three steps.

*Proof.* We put $\Delta' = \inf\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$.

*Step 1.* We want to prove that we have $\Delta' = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$. Indeed, as we have $\underline{b}(1) = 0 \le F(\bar{b}(1))$, the set $\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$ is non-empty. Therefore, we can find a sequence $(\delta_k)$ in $[0, 1]$ verifying:

1. $\forall k, \underline{b}(\delta_k) \le F(\bar{b}(\delta_k))$,

2. $\forall k, \delta_k \ge \Delta'$,

3. $\delta_k \to \Delta'$.
As $\forall k, \bar{b}(\delta_k) \ge \bar{b}(\Delta')$ and $\bar{b}(\delta_k) \to \bar{b}(\Delta')$, the right-continuity of the application $F$ at $\bar{b}(\Delta')$ implies:

$$F(\bar{b}(\delta_k)) \to F(\bar{b}(\Delta')).$$

The inequalities $\underline{b}(\delta_k) \le F(\bar{b}(\delta_k))$ lead, by the passage to the limit, to:

$$\underline{b}(\Delta') \le F(\bar{b}(\Delta')).$$

So $\Delta' = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$.

• Step 2. We want to prove that $\Delta \le \Delta'$.

We have:

$$\underline{b}(\Delta') \le F(\bar{b}(\Delta')) \le \bar{b}(\Delta').$$

By (3) and by noticing that $F(\bar{b}(\Delta')) \in C$, we deduce:

$$\Delta \le \|b - F(\bar{b}(\Delta'))\| \le \Delta'$$

• Step 3. We want to prove that $\Delta' \le \Delta$. For this it is sufficient to show that:

$$\forall c \in C, \Delta' \le \|b - c\|.$$

Indeed, for any $c \in C$, we put $\delta = \|b - c\|$, and we have by (3):

$$\underline{b}(\delta) \le c \le \bar{b}(\delta).$$

Therefore $F(c) = c$ and the growth of $F$ lead to:

$$\underline{b}(\delta) \le c = F(c) \le F(\bar{b}(\delta)).$$

So:

$$\Delta' \le \delta = \|b - c\|.$$

We have proved that $\Delta = \Delta' = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$.

□

We define the set of Chebyshev approximations of $b$:

**Definition 2.** The set of Chebyshev approximations of $b$ is defined using the set $C$, see (13), and the Chebyshev distance associated to $b$ (Definition 1):

$$C_b = \{c \in C \mid \|b - c\| = \Delta(A, b)\}. \qquad (16)$$

We have:

**Corollary 1.** $F(\bar{b}(\Delta)) \in C_b$ is the greatest Chebyshev approximation of the second member $b$ of the system (S).

*Proof.* Let us first show that we have $\|b - F(\bar{b}(\Delta))\| = \Delta$.

Indeed, from (Theorem 1) and (Proposition 1), we know that $\underline{b}(\Delta) \le F(\bar{b}(\Delta)) \le \bar{b}(\Delta)$, so by (3), $\|b - F(\bar{b}(\Delta))\| \le \Delta$.

As $F(\bar{b}(\Delta)) \in C$ (Proposition 1), we know by (Definition 1):

$$\Delta \le \|b - F(\bar{b}(\Delta))\|.$$

We have proven that $F(\bar{b}(\Delta)) \in C_b$.

To prove that $F(\bar{b}(\Delta))$ is the greatest Chebyshev approximation of the second member $b$ of the system (S) we must prove that $c \le F(\bar{b}(\Delta))$ for any $c \in C_b$.

Let $c \in C_b$. As $\Delta = \|b - c\|$, by (3) we have:

$$\underline{b}(\Delta) \le c \le \bar{b}(\Delta).$$

By the growth of $F$, we deduce:

$$c = F(c) \le F(\bar{b}(\Delta)).$$

□

As a consequence, we have:

**Corollary 2.**

$$\Delta = \min_{c \in C} \|b - c\|.$$

$\Delta = 0 \iff$ the system (S) is consistent.

Therefore, $\Delta = 0$ is a necessary and sufficient condition for the system (S) to be consistent.
### 3 Preliminaries computations

In this section, we solve two inequalities. The first one will be involved in the analytical formula to compute the Chebyshev distance associated to the second member of a system of max-T fuzzy relational equations when the t-norm is the usual product i.e., $T = T_P$, see (4). The second one will be involved in the formula to compute the Chebyshev distance when we use Lukasiewicz's t-norm i.e., $T = T_L$, see (5). A similar study was carried out in (Baaj, 2023) for max-min composition.

#### 3.1 T-norm product $T = T_P$

Let $x, y, z, u \in [0, 1]$ be fixed. For the t-norm product $T_P$, let us study the following inequality that involves the Goguen product $\xrightarrow{GG}$ (see (4)), for $\delta \in [0, 1]$:

$$ \underline{x}(\delta) \le u \cdot (y \xrightarrow{GG} \bar{z}(\delta)) \qquad (17) $$

$$ \text{where } y \xrightarrow{GG} \bar{z}(\delta) = \begin{cases} 1 & \text{if } y - z \le \delta \\ \frac{z+\delta}{y} & \text{if } y - z > \delta \end{cases}. $$

We look for the smallest value of $\delta$ so that the inequality (17) is true.

Let:

$$ \sigma_{GG}(u, x, y, z) = \max[(x - u)^{+}, \min(\varphi(u, x, y, z), (y - z)^{+})] \qquad (18) $$

$$ \text{where } \varphi(u, x, y, z) = \begin{cases} \frac{(x \cdot y - u \cdot z)^{+}}{u+y} & \text{if } u > 0 \\ x & \text{if } u = 0 \end{cases}. $$

**Proposition 2.** For all $\delta \in [0, 1]$, we have:

$$ \underline{x}(\delta) \le u \cdot (y \xrightarrow{GG} \bar{z}(\delta)) \iff \sigma_{GG}(u, x, y, z) \le \delta. $$

*Proof.*

If $u = 0$, then $\sigma_{GG}(0, x, y, z) = x$ and we immediately get the desired equivalence:

$$ (x - \delta)^{+} \le 0 \iff x \le \delta \iff \sigma_{GG}(0, x, y, z) \le \delta. $$

It remains for us to study the case where $u > 0$.

$$ \text{Suppose that } (x - \delta)^{+} \le u(y \xrightarrow{GG} \bar{z}(\delta)). $$

* If $y - z \le \delta$, we have:

$$ (x - \delta)^{+} \le u(y \xrightarrow{GG} \bar{z}(\delta)) = u \cdot 1 = u. $$

We know that the inequality $(x - \delta)^{+} \le u$ is equivalent to the inequality $(x - u)^{+} \le \delta$ (Baaj, 2023). As $\min(\varphi(u, x, y, z), (y - z)^{+}) \le (y - z)^{+} \le \delta$, we obtain:

$$ \sigma_{GG}(u, x, y, z) = \max[(x - u)^{+}, \min(\varphi(u, x, y, z), (y - z)^{+})] \le \delta. $$

* If $y - z > \delta$, we have:

$$ (x - \delta)^{+} \le u(y \xrightarrow{GG} \bar{z}(\delta)) = u \frac{z + \delta}{y} \le u $$

so, $(x - u)^{+} \le \delta$ and moreover

$$ \begin{aligned} (x - \delta)^{+} - u(y \xrightarrow{GG} \bar{z}(\delta)) &= \max(x - \delta - \frac{uz + u\delta}{y}, -\frac{uz + u\delta}{y}) \\ &= \max(\frac{xy - \delta y - uz - u\delta}{y}, -\frac{uz + u\delta}{y}) \le 0. \end{aligned} $$

The last inequality is equivalent to $xy - \delta y - uz - u\delta \le 0$ and also:

$$ \frac{xy - uz}{u + y} \le \delta. $$
Then:

• If $(xy - uz)^+ = 0$, we have:

$$ \varphi(u, x, y, z) = \frac{(xy - uz)^+}{u + y} = 0 $$

and also

$$ \sigma_{GG}(u, x, y, z) = \max[(x - u)^+, \min(\varphi(u, x, y, z), (y - z)^+)] = (x - u)^+ \leq \delta. $$

• If $(xy - uz)^+ > 0$, we have:

$$ \varphi(u, x, y, z) = \frac{xy - uz}{u + y} \leq \delta $$

and also

$$ \begin{aligned} \sigma_{GG}(u, x, y, z) &= \max[(x - u)^+, \min(\varphi(u, x, y, z), (y - z)^+)] \\ &= \max[(x - u)^+, \varphi(u, x, y, z)] \\ &\leq \delta. \end{aligned} $$

Let us prove now the remaining implication.

Suppose that $\sigma_{GG}(u, x, y, z) \leq \delta$:

• If $(x - \delta)^+ = 0$, we get trivially that $(x - \delta)^+ = 0 \leq u(y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta))$.

• If $y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta) = 1$, we must prove that $(x - \delta)^+ \leq u$, which is equivalent to $(x - u)^+ \leq \delta$. We have:

$$ (x - u)^+ \leq \max[(x - u)^+, \min(\varphi(u, x, y, z), (y - z)^+)] = \sigma_{GG}(u, x, y, z) \leq \delta. $$

It remains for us to study the case where $(x - \delta)^+ = x - \delta > 0$ and $y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta) < 1$.

From $y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta) < 1$, we deduce:

$$ y - z > \delta \quad \text{and} \quad y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta) = \frac{z + \delta}{y}. $$

Then, we have:

$$ \begin{aligned} (x - \delta)^+ - u(y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta)) &= x - \delta - \frac{uz + u\delta}{y} \\ &= \frac{u + y}{y} \left( \frac{xy - uz}{u + y} - \delta \right). \end{aligned} $$

To obtain $(x - \delta)^+ - u(y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta)) \leq 0$, it is sufficient to prove that

$$ \frac{xy - uz}{u + y} \leq \delta. $$

• If $xy \leq uz$, we have $\frac{xy - uz}{u + y} \leq 0 \leq \delta$.

• If $xy > uz$ and taking into account the inequality $y - z > \delta$, we deduce:

$$ \delta \geq \sigma_{GG}(u, x, y, z) \geq \min\left(\frac{(xy - uz)^+}{u + y}, y - z\right) = \frac{xy - uz}{u + y}. $$

□

From (Proposition 2), we immediately deduce:

**Theorem 2.** For any $x, y, z, u \in [0, 1]$, we have:

$$ \sigma_{GG}(u, x, y, z) = \min\{\delta \in [0, 1] \mid \underline{x}(\delta) \leq u \cdot (y \underset{GG}{\stackrel{\longrightarrow}{z}} \bar{z}(\delta))\}. \quad (19) $$
We illustrate this result:

**Example 3.** Let $x = 0.4$, $y = 0.3$, $z = 0.6$ and $u = 0.2$. We want to obtain the smallest value of $\delta \in [0, 1]$ so that $\underline{x}(\delta) \le u \cdot (y \xrightarrow{GG} \bar{z}(\delta))$ is true. We have $u \cdot (y \xrightarrow{GG} z) = 0.2$.

We compute $\varphi(u, x, y, z) = \frac{(0.4 \cdot 0.3 - 0.2 \cdot 0.6)^+}{0.2 + 0.3} = 0$.

$$ 
\begin{aligned}
\delta &= \sigma_{GG}(u, x, y, z) \\
&= \max[(0.4 - 0.2)^+, \min(0, (0.3 - 0.6)^+)] \\
&= 0.2.
\end{aligned}
$$

We have $\underline{x}(\delta) = x - 0.2 = 0.2$ and $\bar{z}(\delta) = z + 0.2 = 0.8$. Therefore, $u \cdot (y \xrightarrow{GG} \bar{z}(\delta)) = 0.2$ and:

$$ \underline{x}(\delta) = u \cdot (y \xrightarrow{GG} \bar{z}(\delta)). $$

## 3.2 Lukasiewicz's t-norm $T = T_L$

For Lukasiewicz's t-norm $T_L$, we study the following inequality that involves Lukasiewicz's implication $\xrightarrow{L}$ (see (5)) for $\delta \in [0, 1]$:

$$ \underline{x}(\delta) \le \max(0, y \xrightarrow{L} \bar{z}(\delta) - u), \qquad (20) $$

where $y \xrightarrow{L} \bar{z}(\delta) = \begin{cases} 1 & \text{if } y - z \le \delta \\ 1 - y + z + \delta & \text{if } y - z > \delta \end{cases}$

Let $v = x + u - 1$ and:

$$ \sigma_L(u, x, y, z) = \min(x, \max(v^+, \frac{(v + y - z)^+}{2})). \qquad (21) $$

Then:

**Proposition 3.** For all $\delta \in [0, 1]$, we have:

$$ \underline{x}(\delta) \le \max(0, y \xrightarrow{L} \bar{z}(\delta) - u) \iff \sigma_L(u, x, y, z) \le \delta. \qquad (22) $$

*Proof.* We remark that for $x \le \delta$ the two assertions of the equivalence are verified: we have $(x - \delta)^+ = 0$ and $\sigma_L(u, x, y, z) \le x \le \delta$. It remains for us to show the equivalence in the case where $x > \delta$.

Set $f(u, x) = (x + u - 1)^+$, $g(u, x, y, z) = \frac{(x + u - 1 + y - z)^+}{2}$. Then, we have:

$$ \sigma_L(u, x, y, z) = \min(x, \max(f(u, x), g(u, x, y, z))). $$

Let us prove that $\underline{x}(\delta) \le \max(0, y \xrightarrow{L} \bar{z}(\delta) - u) \implies \sigma_L(u, x, y, z) \le \delta$.

As we suppose that $x > \delta$, we must prove:

$$ \max(f(u, x), g(u, x, y, z)) \le \delta. $$

This last inequality will be proven in the following two steps:

1. If $y - z \le \delta$, then $y \xrightarrow{L} \bar{z}(\delta) - u = 1 - u$ and we have:

$$ x - \delta \le \max(0, 1 - u) $$

which we put in the form

$$ 
\begin{aligned}
x - \delta - \max(0, 1 - u) &= x - \delta + \min(0, u - 1) \\
&= \min(x - \delta, x + u - 1 - \delta) \\
&\le 0.
\end{aligned}
$$

As $x - \delta > 0$, we deduce that $x + u - 1 \le \delta$ and also:

$$ f(u, x) = (x + u - 1)^+ \le \delta. $$
Let us show that we also have:

$$g(u, x, y, z) = \frac{(x + u - 1 + y - z)^{+}}{2} \leq \delta.$$

Indeed, if $g(u, x, y, z) = 0$, we have trivially $g(u, x, y, z) = 0 \leq \delta$.

If $g(u, x, y, z) > 0$, then we have:

$$ \begin{aligned} g(u, x, y, z) &= \frac{x + u - 1 + y - z}{2} \\ &\leq \frac{x + u - 1 + \delta}{2} \\ &= \frac{x + u - 1}{2} + \frac{\delta}{2} \\ &\leq \frac{(x + u - 1)^{+}}{2} + \frac{\delta}{2} \quad (\text{we already know that } f(u, x) = (x + u - 1)^{+} \leq \delta) \\ &\leq \frac{\delta}{2} + \frac{\delta}{2} = \delta. \end{aligned} $$

In summary, we have shown $y - z \leq \delta \implies \max(f(u, x), g(u, x, y, z)) \leq \delta$.

2. If $y - z > \delta$, then $y \xrightarrow{L} \bar{z}(\delta) - u = 1 - y + z + \delta - u$ and we have:

$$x - \delta \leq \max(0, 1 - y + z + \delta - u).$$

We rewrite this last inequality in the form:

$$ \begin{aligned} x - \delta - \max(0, 1 - y + z + \delta - u) &= x - \delta + \min(0, u - 1 + y - z - \delta) \\ &= \min(x - \delta, u - 1 + y - z - 2\delta) \\ &\leq 0. \end{aligned} $$

As $x - \delta > 0$, we deduce $x + u - 1 + y - z \leq 2\delta$ and also:

$$g(u, x, y, z) = \frac{(x + y - z + u - 1)^{+}}{2} \leq \delta.$$

It remains for us to show that $f(u, x) = (x + u - 1)^{+} \leq \delta$.

In fact, if $f(u, x) > 0$, then we have:

$$ \begin{aligned} \frac{x + u - 1 + y - z}{2} &> \frac{x + u - 1 + \delta}{2} \\ &= \frac{x + u - 1}{2} + \frac{\delta}{2} \\ &= \frac{f(u, x)}{2} + \frac{\delta}{2}. \end{aligned} $$

We deduce that $x + u - 1 + y - z > 0$, and also:

$$g(u, x, y, z) = \frac{x + u - 1 + y - z}{2} > \frac{f(u, x)}{2} + \frac{\delta}{2}.$$

As we have already shown that $g(u, x, y, z) \leq \delta$, we finally obtain:

$$f(u, x) < 2g(u, x, y, z) - \delta \leq 2\delta - \delta = \delta.$$

In summary, we have shown $y - z > \delta \implies \max(f(u, x), g(u, x, y, z)) \leq \delta$.

We have completed the proof of the implication $\implies$.

Let us assume now that:

$$\sigma_L(u, x, y, z) = \min(x, \max((x + u - 1)^{+}, \frac{(x + u - 1 + y - z)^{+}}{2})) \leq \delta$$

and let us show that:

$$(x - \delta)^{+} \leq \max(0, y \xrightarrow{L} \bar{z}(\delta) - u).$$
As we suppose that $x > \delta$, the inequality $\sigma_L(u, x, y, z) \le \delta$ becomes:

$$ \max(f(u, x), g(u, x, y, z)) = \max((x+u-1)^+, \frac{(x+u-1+y-z)^+}{2}) \le \delta. $$

Let us put the inequality to be shown in the form:

$$ (x - \delta)^+ - \max(0, y \xrightarrow{L} \bar{z}(\delta) - u) \le 0 $$

and distinguish the two cases: $y - z \le \delta$ and $y - z > \delta$.

1. If $y - z \le \delta$, the inequality to be shown becomes:

$$ \begin{aligned} x - \delta - \max(0, 1 - u) &= x - \delta + \min(0, u - 1) \\ &= \min(x - \delta, x + u - 1 - \delta) \\ &\le 0. \end{aligned} $$

As $x - \delta > 0$, we must show that $x + u - 1 - \delta \le 0$, i.e., $x + u - 1 \le \delta$.

But, we have:

$$ x + u - 1 \le (x + u - 1)^+ = f(u, x) \le \max(f(u, x), g(u, x, y, z)) \le \delta. $$

2. If $y - z > \delta$, the inequality to be shown becomes:

$$ \begin{aligned} x - \delta - \max(0, 1 - y + z + \delta - u) &= x - \delta + \min(0, u - 1 + y - z - \delta) \\ &= \min(x - \delta, x + u - 1 + y - z - 2\delta) \\ &\le 0. \end{aligned} $$

So, we must show that $x + u - 1 + y - z - 2\delta \le 0$, i.e. $\frac{x + u - 1 + y - z}{2} \le \delta$.

But, we have:

$$ \frac{x + u - 1 + y - z}{2} \le \frac{(x + u - 1 + y - z)^+}{2} = g(u, x, y, z) \le \max(f(u, x), g(u, x, y, z)) \le \delta. $$

We have completed the proof of implication $\Leftrightarrow$.

□

From (Proposition 3), we immediately deduce:

**Theorem 3.** For any $x, y, z, u \in [0, 1]$, we have:

$$ \sigma_L(u, x, y, z) = \min\{\delta \in [0, 1] \mid \underline{x}(\delta) \le \max(0, y \xrightarrow{L} \bar{z}(\delta) - u)\}. \quad (23) $$

We illustrate this result:

**Example 4.** Let $x = 0.4$, $y = 0.6$, $z = 0.3$ and $u = 0.6$. We want to obtain the smallest value of $\delta \in [0, 1]$ so that $\underline{x}(\delta) \le \max(0, y \xrightarrow{L} \bar{z}(\delta) - u)$ is true.

We have $\max(0, y \xrightarrow{L} z - u) = 0.1$ and $v = x + u - 1 = 0$.

$$ \begin{aligned} \delta &= \sigma_L(u, x, y, z) \\ &= \min(0.4, \max(0, \frac{(0 + 0.6 - 0.3)^+}{2})) \\ &= 0.15. \end{aligned} $$

We have $\underline{x}(\delta) = x - 0.15 = 0.25$ and $\bar{z}(\delta) = z + 0.15 = 0.45$. Therefore, $\max(0, y \xrightarrow{L} \bar{z}(\delta) - u) = 0.25$ and:

$$ \underline{x}(\delta) = \max(0, y \xrightarrow{L} \bar{z}(\delta) - u). $$
## 4 Analytical formulas to compute the Chebyshev distances associated to the second members of the systems $A\Box_{T_P}^{\max}x = b$ and $A\Box_{T_L}^{\max}x = b$

In this section, we compute the Chebyshev distance $\Delta$ (Definition 1) associated to the second member of a system of max-product fuzzy relational equations $A\Box_{T_P}^{\max}x = b$, by an explicit analytical formula in terms of components of the matrix $A$ and those of the second member $b$. We also give an analytical formula for the Chebyshev distance associated to the second member of a system of max-Lukasiewicz fuzzy relational equations $A\Box_{T_L}^{\max}x = b$.

We rely on the equality: $\Delta = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$, see (15), which was proven for any continuous t-norm.

**Notation 1.** Using a continuous t-norm $T$ whose associated implication operator is $\mathcal{I}_T$, we set for $1 \le i, k \le n$ and $1 \le j \le m$:

* $K_i = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le F(\bar{b}(\delta))_i\}$,

* $\beta_j = \min_{1 \le k \le n} \mathcal{I}_T(a_{kj}, \bar{b}(\delta))_k$,

* $W_{ij}^T = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le T(a_{ij}, \beta_j)\}$.

### 4.1 Analytical formula to compute the Chebyshev distance associated to the second member $b$ of the system $A\Box_{T_P}^{\max}x = b$

For any $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$, we have in this case:

$$W_{ij}^{T_P} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le T_P(a_{ij}, \beta_j)\} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le a_{ij}\beta_j\}.$$

where $\beta_j = \min_{1 \le k \le n} \mathcal{I}_{T_P}(a_{kj}, \bar{b}(\delta)_k) = \min_{1 \le k \le n} a_{kj} \underset{GG}{\longrightarrow} \bar{b}(\delta)_k$. Then, we have:

$$W_{ij}^{T_P} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le \min_{1 \le k \le n} a_{kj} \underset{GG}{\longrightarrow} \bar{b}(\delta)_k\}. \quad (24)$$

As $F(\bar{b}(\delta))_i = \max_{1 \le j \le m} T_P(a_{ij}, \beta_j)$, we still have:

$$K_i = \bigcup_{1 \le j \le m} W_{ij}^{T_P}. \quad (25)$$

The Chebyshev distance $\Delta$ associated to the second member $b$ of the system of max-product fuzzy relational equations $A\Box_{T_P}^{\max}x = b$ is given by the following formula:

**Theorem 4.**

$$\Delta = \max_{1 \le i \le n} \delta_i \quad (26)$$

where for $i = 1, 2, \dots, n$:

$$\delta_i = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{ij}, b_i, a_{kj}, b_k). \quad (27)$$

See (18) for the definition of $\sigma_{GG}$.

*Proof.* For any $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$, we deduce from (24) that for $\delta \in [0, 1]$, we have:

$$\delta \in W_{ij}^{T_P} \iff \delta \ge \max_{1 \le k \le n} \sigma_{GG}(a_{ij}, b_i, a_{kj}, b_k).$$

Using (25), we get:

$$\delta \in K_i \iff \exists j \in \{1, 2, \dots, m\} \text{ such that } \delta \ge \max_{1 \le k \le n} \sigma_{GG}(a_{ij}, b_i, a_{kj}, b_k).$$

So, we obtain:

$$\delta \in K_i \iff \delta \ge \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{ij}, b_i, a_{kj}, b_k).$$

As by definition $\delta \in K_i \iff \underline{b}(\delta)_i \le F(\bar{b}(\delta))_i$ and $\Delta = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$, we get:

$$\Delta = \max_{1 \le i \le n} \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{ij}, b_i, a_{kj}, b_k).$$

□
The following example illustrates this result:

**Example 5.** In what follows, we rely on the following matrix and vector, which were used as example in (Cuninghame-Green and Cechlárová, 1995; Pedrycz, 1990b):

$$A = \begin{bmatrix} 1 & 0.4 & 0.5 & 0.7 \\ 0.7 & 0.5 & 0.3 & 0.5 \\ 0.2 & 1 & 1 & 0.6 \\ 0.4 & 0.5 & 0.5 & 0.8 \end{bmatrix} \quad \text{and} \quad b = \begin{bmatrix} 0.4 \\ 1 \\ 0.2 \\ 0 \end{bmatrix}. \qquad (28)$$

For the system $A\square_{T_P}^{\max}x = b$ which uses the t-norm product $T_P$, we compute from (Theorem 4):

$$\delta_1 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{1j}, b_1, a_{kj}, b_k) = \min(0.11, 0.23, 0.2, 0.21) = 0.11,$$

$$\delta_2 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{2j}, b_2, a_{kj}, b_k) = \min(0.42, 0.6, 0.72, 0.62) = 0.42,$$

$$\delta_3 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{3j}, b_3, a_{kj}, b_k) = \min(0.13, 0.07, 0.07, 0.11) = 0.07,$$

$$\delta_4 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{GG}(a_{4j}, b_4, a_{kj}, b_k) = \min(0.0, 0.0, 0.0, 0.0) = 0.0.$$

Therefore, the Chebyshev distance associated to the second member b of the system $A\square_{T_P}^{\max}x = b$ is $\Delta = \max(\delta_1, \delta_2, \delta_3, \delta_4) = 0.42$.

We compute $\bar{b}(\Delta) = \begin{bmatrix} 0.82 \\ 1 \\ 0.62 \\ 0.42 \end{bmatrix}$ and $F(\bar{b}(\Delta)) = \begin{bmatrix} 0.82 \\ 0.57 \\ 0.62 \\ 0.42 \end{bmatrix}$ is the greatest Chebyshev approximation of the second member b.

## 4.2 Analytical formula to compute the Chebyshev distance associated to the second member b of the system

$$A\square_{T_L}^{\max}x = b$$

For any $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$, we have in this case:

$$W_{ij}^{T_L} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le T_L(a_{ij}, \beta_j)\} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le \max(0, a_{ij} + \beta_j - 1)\}.$$

$$\text{where } \beta_j = \min_{1 \le k \le n} \mathcal{I}_{T_L}(a_{kj}, \bar{b}(\delta)_k) = \min_{1 \le k \le n} a_{kj} \underset{L}{\longrightarrow} \bar{b}(\delta)_k.$$

Set $u_{ij} = 1 - a_{ij}$. Then, we have:

$$\begin{aligned} T_L(a_{ij}, \beta_j) &= \max(0, \beta_j - u_{ij}) \\ &= \max(0, \min_{1 \le k \le n} a_{kj} \underset{L}{\longrightarrow} \bar{b}(\delta)_k] - u_{ij}) \\ &= \min_{1 \le k \le n} \max(0, [a_{kj} \underset{L}{\longrightarrow} \bar{b}(\delta)_k] - u_{ij}) \\ &= \min_{1 \le k \le n} ([a_{kj} \underset{L}{\longrightarrow} \bar{b}(\delta)_k] - u_{ij})^+ . \end{aligned}$$

So, we obtain:

$$W_{ij}^{T_L} = \{\delta \in [0, 1] \mid \underline{b}(\delta)_i \le \min_{1 \le k \le n} ([a_{kj} \underset{L}{\longrightarrow} \bar{b}(\delta)_k] - u_{ij})^+\}. \qquad (29)$$

As $F(\bar{b}(\delta))_i = \max_{1 \le j \le m} T_L(a_{ij}, \beta_j)$, we still have:

$$K_i = \bigcup_{1 \le j \le m} W_{ij}^{T_L}. \qquad (30)$$

The Chebyshev distance $\Delta$ associated to the second member b of the system of max-Lukasiewicz fuzzy relational equations $A\square_{T_L}^{\max}x = b$ is given by the following formula:
**Theorem 5.**

$$ \Delta = \max_{1 \le i \le n} \delta_i \qquad (31) $$

where for any $i = 1, 2, \dots, n$:

$$ \delta_i = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_L (1 - a_{ij}, b_i, a_{kj}, b_k). \qquad (32) $$

See (21) for the definition of $\sigma_L$.

*Proof.* For any $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$, we deduce from (29) that for $\delta \in [0, 1]$, we have:

$$ \delta \in W_{ij}^{T_L} \iff \delta \ge \max_{1 \le k \le n} \sigma_{LK} (1 - a_{ij}, b_i, a_{kj}, b_k). $$

Using (30), we get:

$$ \delta \in K_i \iff \exists j \in \{1, 2, \dots, m\} \text{ such that } \delta \ge \max_{1 \le k \le n} \sigma_{LK} (1 - a_{ij}, b_i, a_{kj}, b_k). $$

So, we obtain:

$$ \delta \in K_i \iff \delta \ge \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{LK} (1 - a_{ij}, b_i, a_{kj}, b_k). $$

As by definition $\delta \in K_i \iff \underline{b}(\delta)_i \le F(\bar{b}(\delta))_i$ and $\Delta = \min\{\delta \in [0, 1] \mid \underline{b}(\delta) \le F(\bar{b}(\delta))\}$, we get:

$$ \Delta = \max_{1 \le i \le n} \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_{LK} (1 - a_{ij}, b_i, a_{kj}, b_k). $$

□

**Example 6.** We rely on the matrix A and the vector b used in the previous example, see (28), to construct the system $A \square_{T_L}^{\max} x = b$, which uses with the Lukasiewicz's t-norm $T_L$. From (Theorem 5), we have:

$$ \delta_1 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_L (1 - a_{1j}, b_1, a_{kj}, b_k) = \min(0, 0.4, 0.35, 0.25) = 0, $$

$$ \delta_2 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_L (1 - a_{2j}, b_2, a_{kj}, b_k) = \min(0.45, 0.65, 0.75, 0.65) = 0.45, $$

$$ \delta_3 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_L (1 - a_{3j}, b_3, a_{kj}, b_k) = \min(0.2, 0, 0, 0.2) = 0, $$

$$ \delta_4 = \min_{1 \le j \le m} \max_{1 \le k \le n} \sigma_L (1 - a_{4j}, b_4, a_{kj}, b_k) = \min(0, 0, 0, 0) = 0. $$

Therefore, the Chebyshev distance associated to the second member b of the system $A \square_{T_L}^{\max} x = b$ is $\Delta = \max(\delta_1, \delta_2, \delta_3, \delta_4) = 0.45$.

We compute $\bar{b}(\Delta) = \begin{bmatrix} 0.85 \\ 1 \\ 0.65 \\ 0.45 \end{bmatrix}$ and $F(\bar{b}(\Delta)) = \begin{bmatrix} 0.85 \\ 0.55 \\ 0.65 \\ 0.45 \end{bmatrix}$ is the greatest Chebyshev approximation of the second member b.

# 5 Conclusion

In this article, we gave explicit analytical formulas for computing the Chebyshev distance associated to the second member of a system of max-product fuzzy relational equations and that associated to the second member of a system of max-Lukasiewicz fuzzy relational equations.

The results of (Baaj, 2023) may be then extended to these two systems. For each system, we can describe the sets of Chebyshev approximations of its second member and its approximate solutions set. Similarly to the max-min learning paradigm introduced in (Baaj, 2023), in which the learning error is expressed in terms of $L_\infty$ norm, we can tackle the development of a max-product (resp. max-Lukasiewicz) learning paradigm. The formula of (Baaj, 2023), which computes the minimum value of the learning error according to training data and the method to construct approximate weight matrices whose learning error is minimal can be directly extended to the case of the max-product composition or the max-Lukasiewicz composition.

As applications, we have the complete solution of the problem of the invertibility of a fuzzy relation for max-Lukasiewicz composition: we know the set of matrices A which admit a pre-inverse or a post-inverse. Moreover, for max-min composition, if a matrix A has no pre-inverse (resp. post-inverse), we know how to compute, using the $L_\infty$ norm, an approximate preinverse (resp. postinverse) for A. For both cases of the max-min composition and the max-product composition, the problem of the invertibility of a fuzzy relation is already solved (Wu et al., 2021, 2022).
References

Ismaël Baaj. Max-min learning of approximate weight matrices from fuzzy data, 2023. URL https://arxiv.org/abs/2301.06141.

RA Cuninghame-Green and Katarína Cechlárová. Residuation in fuzzy algebra and some applications. *Fuzzy Sets and Systems*, 71(2):227–239, 1995.

Antonio Di Nola, Witold Pedrycz, and Salvatore Sessa. On solution of fuzzy relational equations and their characterization. *Busefal*, 12:60–71, 1982.

Antonio Di Nola, Witold Pedrycz, Salvatore Sessa, and Wang Pei Zhuang. Fuzzy relation equation under a class of triangular norms: A survey and new results. *Stochastica*, 8(2):99–145, 1984.

Antonio Di Nola, Elie Sanchez, Witold Pedrycz, and Salvatore Sessa. *Fuzzy relation equations and their applications to knowledge engineering*. Kluwer Academic Publishers, 1989.

Siegfried Gottwald. Characterizations of the solvability of fuzzy equations. *Elektronische Informationsverarbeitung und Kybernetik*, 22(2/3):67–91, 1986.

Kaoru Hirota and Witold Pedrycz. Fuzzy relational compression. *IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics)*, 29(3):407–415, 1999.

Erich Peter Klement, Radko Mesiar, and Endre Pap. *Triangular norms*, volume 8. Springer Science & Business Media, 2013.

George J Klir and Bo Yuan. Approximate solutions of systems of fuzzy relation equations. In *Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference*, pages 1452–1457. IEEE, 1994.

Pingke Li and Shu-Cherng Fang. Chebyshev approximation of inconsistent fuzzy relational equations with max-t composition. In *Fuzzy Optimization*, pages 109–124. Springer, 2010.

Masaaki Miyakoshi and Masaru Shimbo. Solutions of composite fuzzy relational equations with triangular norms. *Fuzzy Sets and Systems*, 16(1):53–63, 1985. ISSN 0165-0114.

Hajime Nobuhara, Witold Pedrycz, Salvatore Sessa, and Kaoru Hirota. A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations. *Information Sciences*, 176(17):2526–2552, 2006.

Witold Pedrycz. Fuzzy relational equations with triangular norms and their resolutions. *Busefal*, 11:24–32, 1982.

Witold Pedrycz. Numerical and applicational aspects of fuzzy relational equations. *Fuzzy sets and systems*, 11(1):1–18, 1983.

Witold Pedrycz. On generalized fuzzy relational equations and their applications. *Journal of mathematical Analysis and applications*, 107(2):520–536, 1985.

Witold Pedrycz. Algorithms for solving fuzzy relational equations in a probabilistic setting. *Fuzzy Sets and Systems*, 38(3):313–327, 1990a.

Witold Pedrycz. Inverse problem in fuzzy relational equations. *Fuzzy Sets and systems*, 36(2):277–291, 1990b.

Elie Sanchez. Resolution of composite fuzzy relation equations. *Information and control*, 30(1):38–48, 1976.

Elie Sanchez. Solutions in composite fuzzy relation equations: Application to medical diagnosis in brouwerian logic. In M. M. Gupta, G. N. Saridis, and B. R. Gaines, editors, *Fuzzy automata and decision processes*, pages 221–234. Amsterdam: North-Holland, 1977.

Wu Wangming. Fuzzy reasoning and fuzzy relational equations. *Fuzzy sets and systems*, 20(1):67–78, 1986.

Ching-Feng Wen, Yan-Kuen Wu, and Zhaowen Li. Algebraic formulae for solving systems of max-min inverse fuzzy relational equations. *Information Sciences*, 2022. ISSN 0020-0255.

Yan-Kuen Wu, Yung-Yih Lur, Hsun-Chih Kuo, and Chingfeng Wen. An analytical method to compute the approximate inverses of a fuzzy matrix with max-product composition. *IEEE Transactions on Fuzzy Systems*, 2021.

Yan-Kuen Wu, Yung-Yih Lur, Ching-Feng Wen, and Shie-Jue Lee. Analytical method for solving max-min inverse fuzzy relation. *Fuzzy Sets and Systems*, 440:21–41, 2022.

Gang Xiao, Tianxiang Zhu, Yanhua Chen, and Xiaopeng Yang. Linear searching method for solving approximate solution to system of max-min fuzzy relation equations with application in the instructional information resources allocation. *IEEE Access*, 7:65019–65028, 2019.