# On an extension of Girard algebras

Remigijus Petras GYLYS (MII)

e-mail: gyliene@ktl.mii.lt

## 1. Introduction

In this note we present a generalization of the canonical extension of Girard algebras recently introduced and studied by U. Höhle and S. Weber [1]–[4]. Our results are submitted without proofs. We are going to detail it in a subsequent paper.

## 2. Girard algebras, MV-algebras and Boolean algebras

A lattice-theoretic definition of a Girard algebra, of a MV-algebra and of a Boolean algebra are the following:

**DEFINITION 2.1.** A Girard algebra $G$ is a bounded lattice (with the join $\lor$, with the meet $\land$, with the least element $0 = \bigwedge G$ and with the greatest element $1 = \bigvee G$) together with commutative semigroup operation $\circ$ with $1$ as unit and $0$ as zero such that

1. $a \circ (b \lor c) = (a \circ b) \lor (a \circ c)$ (distributivity between $\circ$ and $\lor$),

2. There exists a further binary operation $\backslash$: $G \times G \to G$ defined by

$$a \backslash b = \bigvee \{c \in G \mid a \circ c \leqslant b\} \text{ (existence of residuals),}$$

which satisfies the following axiom:

$$(a \backslash 0) \backslash 0 = a \text{ (Involution).}$$

In any Girard algebra the residual complement $\perp$: $G \to G$ and the dual operation $\Join$: $G \times G \to G$ associated with $\circ$ can be defined by

$$a^{\perp} = a \backslash 0 \text{ and } a \Join b = (a^{\perp} \circ b^{\perp})^{\perp}.$$

**DEFINITION 2.2.**

1. A Girard algebra $G$ is called an MV-algebra ("multi-valued") iff the following axiom is satisfied:

$$a \circ (a \backslash b) = a \land b \text{ (Divisibility).}$$

2. An MV-algebra is called a Boolean algebra iff $\circ = \land$.
### 3. I-extension of Girard algebras

In [2] was introduced the “canonical” extension $G_1$ of a Girard algebra $G$, the Girard algebra $G_1 = \{\langle a_1, a_2 \rangle \in G \times G \mid a_1 = a_2\}$, where $G$ was identified with its diagonal $G_\Delta = \{\langle a, a \rangle \mid a \in G\}$. In this note we generalize this canonical extension of $G$ to the I-extension defined as follows.

**DEFINITION 3.1.** Given a Girard algebra $G$ and the set of natural numbers $I := \{1, 2, \dots, n\}$ for $n \ge 2$, by I-extension $G^I$ of $G$ we shall understand the bounded lattice of all isotone functions $a: I \to G$. We write $a \le b$ to mean that $a_i \le b_i$ for all $i = 1, 2, \dots, n$. The lattice-theoretic operations and universal bounds are given by, for all $i \in I$,

$$ (a \wedge b)_i = a_i \wedge b_i, (a \vee b)_i = a_i \vee b_i, 0_i = 0 \text{ and } 1_i = 1. $$

If we identify $G$ with all constant functions of $G^I$, then $G$ is a sublattice of $G^I$.

**THEOREM 3.2.** Let $G$ be a Girard algebra. Then $G^I$ is also a Girard algebra with structure (denoted by the same symbols) given by:

$$ 1. \quad (a \circ b)_i = \bigvee_{j,k=1}^{n} \{a_j \circ b_k \mid j + k = i + 1\}, $$

$$ 2. \quad (a \bvee b)_i = \bigwedge_{j=1}^{n-i+1} a_j \bvee b_{i+j-1}, $$

$$ 3. \quad (a^{\perp})_i = (a_{n-i+1})^{\perp}, $$

$$ 4. \quad (a \uplus b)_i = \bigwedge_{j,k=1}^{n} \{a_j \uplus b_k \mid j + k = n + i\}. $$

**COROLLARY 3.2.** Let $G$ be Girard algebra, and $G^I$ be its I-extension. Then the following assertions are equivalent:

1. $G^I$ is an MV-algebra.

2. $G$ is a Boolean algebra.

### 4. Conditioning and mean value generation

In [1]–[2] were proposed the following axioms for conditioning operators and mean value functions in Girard algebras:

**DEFINITION 4.1.** Let $G$ be a Girard algebra. A binary operation $|: G \times G \to G$ is called a conditioning operator on $G$ iff $|$ satisfies the following axioms:

1. $a|1 = a$,

2. $(b \circ (b \setminus a))|b = a|b,$
3. $a \le b \Rightarrow a|c \le b|c$,

4. $b \le c$ and $c \circ (c \setminus a) \le b \circ (b \setminus a) \Rightarrow a|c \le a|b$

5. $(a|b)^\perp = (a^\perp \circ b)|b$, particularly, $(0|0)^\perp = 0|0$.

LEMMA 4.2 (Lemma 4.3 [2. ]) Every conditioning operator fulfils the following property:

$$b \circ (b \setminus a) \le a|b \le b \setminus a.$$

DEFINITION 4.3. Let G be a Girard algebra. A mean value function is an isotone, idempotent binary (not necessarily commutative) operation on G, i.e., a map C: G × G → G satisfying the following axioms

1. $C_{a,a} = a$ (idempotency),

2. $C_{a,b} \le C_{c,d}$ whenever $a \le c, b \le d$ (isotonicity).

A mean value function C is said to be compatible with the residual complement in G iff C satisfies the following additional condition

$$(C_{a,b})^{\perp} = C_{b^{\perp},a^{\perp}}.$$

PROPOSITION 4.4 (Theorem 4.5 [2. ]) Let G be a Girard algebra and C be a mean value function on G which is compatible with the residual complement in G. Then C induces a conditioning operator $|\text{on } G$ by

$$a|b = C_{b \circ (b \setminus a), b \setminus a}.$$

PROPOSITION 4.4 (Theorem 4.6 [2. ]) Let G be an MV-algebra and $|\text{be a conditioning operator. Then the following assertions are equivalent:}$

1. $|\text{is a mean value based conditioning operator;}$

2. $|\text{satisfies the condition } a \le b \Rightarrow (a \wedge c)|(c \setminus a) \le (b \wedge c)|(c \setminus b).$

5. Conditioning operators and mean value functions on I-extensions of Girard algebras

LEMMA 5.1. Let G be a Girard algebra and $G^I$ be the I-extension of G, where $I = \{1, 2, \dots, n, n+1, \dots, 2n\}$ with $n \ge 1$. Every mean value function B on G induces a mean value function C on $G^I$ by:

$$(C_{a,b})_i = \begin{cases} B_{a_i, a_{i+1} \wedge b_i} & \text{if } i = 1, \dots, n, \\ (B_{(b_i)^\perp, (b_{i-1} \vee a_i)^\perp})^\perp & \text{if } i = n+1, \dots, 2n. \end{cases}$$

Moreover, C on $G^I$ is compatible with the residual complement in $G^I$ and satisfies the further property:

$$(C_{a,b})_i = \begin{cases} a_1 & \text{if } i = 1, \dots, n, \\ b_1 & \text{if } i = n+1, \dots, 2n, \end{cases}$$

whenever a and b are constant functions, $a \le b$, $a = a_1$, $b = b_1$.
### REMARK 5.2 (existence of mean value functions). 
Let $G$ be a Girard algebra. Then the following maps $B^1$ and $B^2$ defined by

$$B_{a,b}^1 = a \text{ and } B_{a,b}^2 = b$$

are mean value functions on $G$. Further the corresponding mean value functions $C^1$ and $C^2$ (in the sense of the preceding lemma) are given by

$$(C_{a,b}^1)_i = \begin{cases} a_i & \text{if } i = 1, \dots, n, \\ b_i & \text{if } i = n+1, \dots, 2n, \end{cases}$$

$$(C_{a,b}^2)_i = \begin{cases} a_{i+1} \wedge b_i & \text{if } i = 1, \dots, n, \\ b_{i-1} \vee a_i & \text{if } i = n+1, \dots, 2n. \end{cases}$$

### PROPOSITION 5.3. 
Let $G^1$ be the 1-extension of a Girard algebra $G$ with $I = \{1, 2, \dots, 2n\}$. Then there exists a mean value function $C$ on $G^1$ which is compatible with the residual complement in $G^1$ and satisfies the condition in the preceding lemma. Further the mean value based conditioning operator $|\cdot|$ corresponding to $C$ satisfies the additional property:

$$a|b = \begin{cases} b_1 \circ (b_1 \setminus a_1), & \text{if } i = 1, \dots, n, \\ b_1 \setminus a_1, & \text{if } i = n+1, \dots, 2n, \end{cases}$$

whenever $a$ and $b$ are constant functions, $a \le b$, $a = a_1$, $b = b_1$.

## 6. Uncertainty measures on MV-algebras

### DEFINITION 6.1. ([2]) 
Let $G$ be an MV-algebra, and $\perp$ and $\oplus$ be the residual complement and the dual operation associated with $\circ$. A map $m: G \to [0, 1]$ is called an uncertainty measure iff $m$ satisfies the following conditions:

1. $m(0) = 0, m(1) = 1$,

2. $a \le b \Rightarrow m(a) \le m(b)$.

An uncertainty measure $m$ is said to be additive iff it satisfies the axiom:

3. $a \circ b = 0 \Rightarrow m(a \oplus b) = m(a) + m(b)$.

Now we are going to present the last important result of this note (generalizing Theorem 6.5 [2]), to establish the existence of additive measure extension in the Boolean case. For this, we need the following observation

### LEMMA 6.2. 
Let $G$ be a Boolean algebra, and $m$ be a probability measure on it. Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ (with $n \ge 2$) be elements of $G$ such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Then for every $i = 1, \dots, n-1$ the equality holds:
$$ m\left( \bigwedge_{j,k=1}^{n} \{a_j \lor b_k \mid j+k=n+i\} \right) \\ = \sum_{j,k=1}^{n} \{m(a_j \lor b_k) \mid j+k=n+i\} - \sum_{j,k=1}^{n} \{m(a_j \lor b_k) \mid j+k=n+i+1\}. $$

Finally, we arrive at

**THEOREM 6.3.** Let $G$ be a Boolean algebra, and $m$ be a probability measure on $G$. Let $G^1$ be the MV-algebra I-extension of $G$. Then $m$ has a unique extension to an additive uncertainty measure $\tilde{m}$ on $G^1$, i.e., there exists a unique additive uncertainty measure $\tilde{m}$ on $G^1$ such that the restriction of $\tilde{m}$ to $G$ coincides with $m$. In particular, $\tilde{m}$ is given by

$$ \tilde{m}(a) = \frac{1}{n} \sum_{i=1}^{n} m(a_i). $$

References

1. U. Höhle, S. Weber, Uncertainty measures, realizations and entropies, in: J. gaoutsias, R.P.S. Mahler, H.T. Nguyen (Eds.), *Random Sets: Theory and Applications*, Springer-Verlag, Heidelberg/Berlin/NewYork (1997), pp. 259–295.

2. U. Höhle, S. Weber, On conditioning operators, in: U. Höhle, S.E. Rodabaugh (Eds.), *Mathematics of Fuzzy Sets*, Kluwer Academic Publishers, Boston/Dordrecht/London (1999), pp. 653–673.

3. S. Weber, Conditioning on MV-algebras and additive measures, Part I, *Fuzzy Sets and Systems*, **92** (1997), 241–250.

4. S. Weber, Conditioning on MV-algebras and additive measures, Further results, in: D. Dubois, H. Prade, E.P. Klement (Eds.), *Fuzzy Sets, Logics and Reasoning about Knowledge*, Kluwer Academic Publishers, Boston/Dordrecht (1999), pp. 175–199.

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