Tsallis Entropy in MV-Algebras

Abstract

We deal with Tsallis entropy in MV-algebraic dynamical systems. We prove that Tsallis entropy is a submeasure and that it is invariant under isomorphisms. We also provide two examples which show that Tsallis entropy allows one to distinguish some non-isomorphic MV-dynamical systems.

1. Introduction

Motivated by Constantino Tsallis’ significant discovery in 1988 regarding the notion of Tsallis entropy, we aim to introduce Tsallis entropy in the context of MV algebras and develop some of its properties. More specifically, the logical entropy and logical mutual information of partitions in a Riesz MV-algebra were studied in [1]. In this paper, we extend some of the results obtained in [1] from logical entropy to Tsallis entropy.

We recall that in information theory, Shannon’s entropy [2] is widely used, as it allows us to measure the amount of information gained from the realization of an experiment. Using Shannon’s entropy, Kolmogorov and Sinai [3,4] defined the entropy of dynamical systems, and they proved that this entropy is invariant under isomorphisms of dynamical systems.

However, as Ellerman noticed, Shannon’s entropy is not suitable for information theory because mutual information between three variables could be negative (however, this important remark went almost unnoticed in the literature). For this reason, Ellerman proposed logical entropy (see [5]), and Tsallis extended Ellerman’s work (see [6]).

In the classical framework, if $P=(p_1,\dots ,p_n)$ is a probability distribution, then its Tsallis entropy of order $\alpha$, where $\alpha \ne 1$, is defined as the following number:

$$T_{\alpha }\left(P\right)={\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{k=1}^n{p}_k^{\alpha }\right).$$

If X is a continuous random variable with a probability density f, then the Tsallis entropy of order $\alpha$, where $\alpha \ne 1$, is defined as the following number:

$$T_{\alpha }\left(X\right)={\displaystyle \frac{1}{\alpha -1}}\left(1-{\int }_{\mathbb{R}}f{\left(x\right)}^{\alpha }dx\right).$$

Tsallis entropies yield the Boltzmann–Gibbs–Shannon entropy

$$S(x)=-\int f(x)logf(x)dx$$

as a limit case for $\alpha \to 1$, as can easily be verified by writing $\alpha =1+\epsilon$ and taking the limit $\epsilon \to 0$ above. Due to the flexibility of Tsallis entropy compared to Boltzmann–Gibbs–Shannon entropy, it has applications in many areas of information theory, physics, chemistry, and technology. Several properties and statistical aspects of Tsallis entropy can be found in [7,8,9,10,11,12,13].

This paper is organized as follows: Section 2 is devoted to the background of this research line. Definitions of entropy, particularly of logical entropy, and some facts are provided in Section 3. In subsequent sections, we focus on the Tsallis entropy of partitions and on the Tsallis entropy of MV-dynamical systems. More precisely, in Section 5 for A, B–two partitions–we introduced the conditional Tsallis entropy of A given B, and we discussed its properties. In Section 6, we prove that the Tsallis entropy is invariant under isomorphisms of MV-dynamical systems, and we offer two examples which show that Tsallis entropy allows one to distinguish some non-isomorphic MV-dynamical systems. Concluding remarks are presented in the final section.

2. Preliminaries and Notation

In this section, we give some basic notation on MV-algebras, states, and MV-algebraic dynamical systems.

 Definition 1. 

An MV-algebra is an algebra $(E;\oplus ;\neg ;0)$ with a binary operation ⊕, a unary operation ¬, and a constant 0 satisfying the following equations:

(MV1) $x\oplus (y\oplus z)=(x\oplus y)\oplus z;$

(MV2) $x\oplus y=y\oplus x;$

(MV3) $x\oplus 0=x;$

(MV4) $\neg \neg x=x;$

(MV5) $x\oplus \neg 0=\neg 0;$

(MV6) $\neg (\neg x\oplus y)\oplus y=\neg (\neg y\oplus x)\oplus x.$

 Remark 1. 

MV-algebras were introduced by Chang in [14].

On each MV-algebra E, we define Constant 1 and the operations ⊖ and ⊙ as follows:

(1) $1:=\neg 0$;

(2) $x\odot y:=\neg (\neg x\oplus \neg y)$;

(3) $x\ominus y:=x\odot \neg y$.

Let G be an ℓ-group. For each $u\in G,u>0$, let

$$[0,u]=\{x\in G:0\le x\le u\},$$

and for each $x,y\in [0,u]$, let $x\oplus y=u\wedge (x+y)$ and $\neg x:=u-x$. It is not hard to see that $\left(\right[0;u];\oplus ;\neg ;0)$ is an MV-algebra, which will be denoted by $\Gamma (G;u)$.

 Definition 2. 

A Riesz MV-algebra is a structure $(E,·,\oplus ,\neg ,0,1)$ where $(E,\oplus ,\neg ,0,1)$ is an MV-algebra, ⊙ is the Łukasiewicz product, and the operation $·:[0,1]\times E\to E$ satisfies the following identities for any $r,q\in [0,1]$ and $x,y\in E$:

(RMV1) $(r·x)\odot (r·y)=0$ and $r·(x\oplus y)=(r·x)\oplus (r·y)$ whenever $x\odot y=0$;

(RMV2) $(r·x)\odot (q·x)=0$ and $(r\oplus q)·x=(r·x)\oplus (q·x)$ whenever $r\odot q=0$;

(RMV3) $r·(q·x)=\left(rq\right)·x$;

(RMV4) $1·x=x.$

 Remark 2. 

Riesz MV-algebras are, up to isomorphism, unit intervals in Riesz spaces with a strong unit. They were introduced in [15].

In the sequel let E be an MV-algebra.

In every MV-algebra E, a partial sum can be defined by $a+b=a\oplus b$ when $a\le \neg b$, and $a+b$ are undefined otherwise.

 Definition 3. 

We define partition of unity in E as any finite sequence $A=(a_1,\dots ,a_p)$ such that $a_1+\dots +a_p=1$. Notice that the sequence A may contain repetitions.

 Definition 4. 

A refinement of a partition $A=(a_1,\dots ,a_p)$ is a partition $B=(b_1,\dots ,b_q)$ such that there is a partition $({\alpha }_1,\dots ,{\alpha }_p)$ of the set $\{1,\dots ,q\}$, such that $a_i={\sum }_{j\in {\alpha }_i}{b}_j$. We write $A\le B$.

 Definition 5. 

An MV-algebra E is σ-complete if all countable subsets have both a supremum and an infimum.

 Definition 6. 

A state s on E is a mapping $s:E\to [0,1]$ such that $s\left(1\right)=1$ and for all $a,b\in E$ with $a\odot b=0$, $s(a\oplus b)=s\left(a\right)+s\left(b\right)$.

 Definition 7 

([16]). By an MV-dynamical system, we mean a triple $(E,s,\tau )$, where E is a σ-complete MV-algebra, τ is an automorphism of E, and $s:E\to [0,1]$ is a state on E such that $s\circ \tau =s.$

The system is called a Riesz MV-dynamical system if E is a Riesz MV-algebra.

We say that two MV-dynamical systems $(E_1,s_1,{\tau }_1)$ and $(E_2,s_2,{\tau }_2)$ are isomorphic if there exists a bijective mapping $f:E_1\to E_2$ satisfying the following conditions:

(i) f is an MV-algebra isomorphism;

(ii) $f\left({\tau }_1\left(a\right)\right)={\tau }_2\left(f\left(a\right)\right)$ for every $a\in E_1$;

(iii) $s_1\left(a\right)=s_2\left(f\left(a\right)\right)$ for every $a\in E_1$.

Two Riesz MV-dynamical systems are isomorphic if f is a Riesz MV-algebra isomorphism.

3. Entropies

In this section, we briefly recall some definitions of entropy. In particular, we recall the definition of logical entropy:

 Definition 8. 

Let s be as in Definition 7. Then, the Shannon entropy of a partition $A=(a_1,\dots ,a_p)$ is the number

$$H\left(A\right)=-\sum _{i=1}^{p}s\left(a_i\right)log\left(s\left(a_i\right)\right).$$

 Proposition 1 

(([17,18])). Let $A_k={\left\{a_j^k\right\}}_{j=1}^{m_k}$ be a tuple of partitions of unity of E, $k=1,\dots ,n$. It can be shown (using the Riesz decomposition property which holds in every MV algebra) that there is at least one common refinement of $A_1,\dots ,A_n$ of the form

$$C=\left\{c_{i_1\dots i_n}\right|1\le i_1\le m_1,\dots ,1\le i_n\le m_n\},$$

such that

$$a_j^k=\sum \left\{c_{i_1\dots i_n}\right|i_k=j\}.$$

Any such partition is called a Riesz refinement of $A_1,\dots ,A_n$.

 Remark 3. 

By [17] on p. 146, in MV-algebras there exists a unique Riesz refinement of two partitions $A,B$ which satisfies

$$(c_{i+i,j}+\dots +c_{m,j})\wedge (c_{i,j+i}+\dots +c_{i,n})=0.$$

In the sequel, we always denote by $A\vee B$ this canonical Riesz refinement.

 Definition 9. 

Let s be as in Definition 7. Then, the logical entropy of a partition $A=(a_1,\dots ,a_p)$ is the number

$$H_L\left(A\right)=\sum _{i=1}^{p}s\left(a_i\right)(1-s\left(a_i\right)).$$

Let $(E,s,\tau )$ be a dynamical system in an MV-algebra E, and let A be a partition in E. Then we define the logical entropy of A relative to E by:

$$H_L\left(A\right)=\underset{n}{lim}{H}_L\left({\vee }_{k=0}^{n-1}{\tau }^k\left(A\right)\right).$$

The logical entropy of $(E,s,\tau )$ is defined as

$$H_L\left(E\right)=sup\left\{H_L\left(A\right)\right|A\mathit{partition}\mathit{of}E\}.$$

 Remark 4. 

We will see that the logical entropy of $(E,s,\tau )$ actually does not depend on τ.

 Remark 5. 

The logical entropy in the classical setting was introduced by Ellerman in [5].

4. Tsallis Entropy of Partitions

In this section, we introduce and study Tsallis entropy of partitions.

 Definition 10. 

Let $A=(a_1,\dots ,a_p)$ be a partition in an MV-algebra E with a state s. Then, we define the Tsallis entropy of order α, where α is a real positive number different from 1, as

$$T_{\alpha }\left(A\right)={\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{k=1}^{p}s{\left(a_k\right)}^{\alpha }\right).$$

The index α describes the deviation of Tsallis entropy from the Shannon entropy.

 Remark 6. 

If we set $\alpha =2$, we obtain the logical entropy, and it is easy to check that $T_{\alpha }$ is a nonnegative number for α less than 1.

 Notation 1. 

Let $A=(a_1,\dots ,a_p)$ and $B=(b_1,\dots ,b_q)$ be two partitions of unity in E. Let us put

$$s\left(a_i\right|b_j):={\displaystyle \frac{s\left(c_{ij}\right)}{s\left(b_j\right)}}$$

if $s\left(b_j\right)>0$, where $c_{ij}$ is the common refinement of A and B; otherwise, we set $s\left(a_i\right|b_j)=0$.

Moreover, let us write, for $\alpha >0$ and $\alpha \ne 1,$ and $x\in [0,1],$

$$l_{\alpha }\left(x\right):={\displaystyle \frac{1}{\alpha -1}}(x-x^{\alpha }).$$

 Remark 7. 

Notice that, since $l_{\alpha }^{\prime \prime }<0$, the function $l_{\alpha }$ is strictly concave.

 Proposition 2. 

If $A=(a_1,\dots ,a_p)$ is a partition of unity and $y\in E$, then

(i) 

${\sum }_{i=1}^{p}s(a_i\wedge y)=s\left(y\right)$;

(ii) 

${\sum }_{i=1}^{p}s\left(a_i\right|y)=1$ whenever $s\left(y\right)>0$.

 Proof. 

Item (i) holds true since we have ${\sum }_{i=1}^p(a_i\wedge y)=y$.

Item (ii) derives from item (i), as ${\sum }_{i=1}^{p}s\left(a_i\right|y)={\displaystyle \frac{1}{s\left(y\right)}}{\sum }_{i=1}^{p}s(a_i\wedge y)=1$. □

 Proposition 3. 

If $A=(a_1,a_2,\dots ,a_p)$ be any partition of unity in E and $\alpha >1$, then

$$0\le T_{\alpha }\left(A\right)\le {\displaystyle \frac{1}{\alpha -1}}(1-p^{1-\alpha }).$$

The equality holds if and only if $s\left(a_i\right)={\displaystyle \frac{1}{p}}$ for $i=1,2,\dots ,p.$

 Proof. 

The inequality $T_{\alpha }\left(A\right)\ge 0$ follows from the non-negativity of function $l_{\alpha }$, so it is sufficient to prove the second assertion. We will use the Jensen inequality. Since the function $l_{\alpha }$ is strictly concave, by applying the Jensen inequality, we have:

$$l_{\alpha }\left({\displaystyle \frac{1}{p}}\sum _{i=1}^{p}s\left(a_i\right)\right)\ge {\displaystyle \frac{1}{p}}\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right)\right)$$

with the equality if and only if $s\left(a_1\right)=s\left(a_2\right)=\dots =s\left(a_p\right)$. Since ${\sum }_{i=1}^{p}s\left(a_i\right)=1$, it follows that

$$T_{\alpha }\left(A\right)=\sum l_{\alpha }\left(s\left(a_i\right)\right)\le p{l}_{\alpha }\left({\displaystyle \frac{1}{p}}\sum _{i=1}^{p}s\left(a_i\right)\right)$$

$$=p{l}_{\alpha }\left({\displaystyle \frac{1}{p}}\right)={\displaystyle \frac{p}{\alpha -1}}\left({\displaystyle \frac{1}{p}}-{({\displaystyle \frac{1}{p}})}^{\alpha }\right)={\displaystyle \frac{1}{\alpha -1}}(1-p^{1-\alpha }).$$

The equality holds if and only if $s\left(a_1\right)=s\left(a_2\right)=\dots =s\left(a_p\right)$, i.e., if and only if $s\left(a_i\right)={\displaystyle \frac{1}{p}}$ for $i=1,2,\dots ,p.$ □

 Proposition 4. 

Let $A,B$ be partitions of unity such that $A\le B$ and $\alpha >1$. Then,

$$T_{\alpha }\left(A\right)\le T_{\alpha }\left(B\right).$$

 Proof. 

Suppose that $A=(a_1,a_2,\dots a_p),B=(b_1,b_2,\dots ,b_q),A\le B$. Then, there exists a partition $\left(\beta \right(1),\beta (2),\dots \beta (k\left)\right)$ of the set $\{1,2,\dots ,q\}$ such that $a_i={\sum }_{j\in \beta \left(i\right)}{b}_j$ for $i=1,2,\dots ,p$. Therefore,

$$s\left(a_i\right)=s\left(\sum _{j\in \beta \left(i\right)}{b}_j\right)=\sum _{j\in \beta \left(i\right)}s\left(b_j\right)$$

for $i=1,2,\dots ,p$. Then,

$$s{\left(a_i\right)}^{\alpha }={\left(\sum _{j\in \beta \left(i\right)}s\left(b_j\right)\right)}^{\alpha }\ge \sum _{j\in \beta \left(i\right)}s{\left(b_j\right)}^{\alpha },$$

for $i=1,2,\dots ,p$. Summing both sides of the above inequality over i, we get

$$\sum _{i=1}^{p}s{\left(a_i\right)}^{\alpha }\ge \sum _{i=1}^p\sum _{j\in \beta \left(i\right)}s{\left(b_j\right)}^{\alpha }=\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }.$$

In this case, we have ${\displaystyle \frac{1}{\alpha -1}}>0$; hence,

$$T_{\alpha }\left(A\right)={\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{i=1}^{p}s{\left(a_i\right)}^{\alpha }\right)\le {\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\right)=T_{\alpha }\left(B\right).$$

□

 Proposition 5. 

Let $A=(a_1,a_2,\dots a_p),B=(b_1,b_2,\dots ,b_q)$ be partitions. Then, for $\alpha >1$, we have

$$\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)\le T_{\alpha }\left(A\right).$$

 Proof. 

Applying the Jensen inequality, we have

$$\sum _{j=1}^{q}s\left(b_j\right)·l_{\alpha }\left(s\left(a_i\right|b_j)\right)\le l_{\alpha }\left(\sum _{j=1}^{q}s\left(b_j\right)·s\left(a_i\right|b_j)\right)$$

$$=l_{\alpha }\left(\sum _{j=1}^{q}s(a_i\wedge b_j)\right)=l_{\alpha }\left(s\left(a_i\right)\right),$$

for $i=1,2,\dots ,p$, and consequently,

$$(\ast )\sum _{j=1}^{q}s\left(b_j\right)\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)\le \sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right)\right)=T_{\alpha }\left(A\right).$$

The assumption that $\alpha >1$ implies the inequality $s{\left(b_j\right)}^{\alpha }\le s\left(b_j\right)$ for $j=1,2\dots ,q$. The function $l_{\alpha }$ is non-negative; therefore, for $j=1,2\dots ,q$, we get

$$s{\left(b_j\right)}^{\alpha }\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)\le s\left(b_j\right)\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)$$

and so

$$\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)\le \sum _{j=1}^{q}s\left(b_j\right)\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right).$$

The last inequality combined with (*) yields the claim. □

In the next theorem, we prove that Tsallis entropy of partitions is a submeasure.

 Theorem 1. 

Let $A=(a_1,a_2,\dots a_p),B=(b_1,b_2,\dots ,b_q)$ be partitions of unity. Then, for $\alpha >1$, one has

$$T_{\alpha }(A\vee B)\le T_{\alpha }\left(A\right)+T_{\alpha }\left(B\right).$$

 Proof. 

We have

$$T_{\alpha }(A\vee B)={\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{i=1}^p\sum _{j=1}^{q}s{(a_i\wedge b_j)}^{\alpha }\right)=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^{p}s{\left(a_i\right|b_j)}^{\alpha }\right)$$

$$={\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }+\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^{p}s{\left(a_i\right|b_j)}^{\alpha }\right)=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\right)+{\displaystyle \frac{1}{\alpha -1}}\left(\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^{p}s{\left(a_i\right|b_j)}^{\alpha }\right)=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\right)+{\displaystyle \frac{1}{\alpha -1}}\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\left(1-\sum _{i=1}^{p}s{\left(a_i\right|b_j)}^{\alpha }\right)=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\right)+\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }{\displaystyle \frac{1}{\alpha -1}}\left(\sum _{i=1}^{p}s\left(a_i\right|b_j)-\sum _{i=1}^{p}s{\left(a_i\right|b_j)}^{\alpha }\right)=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(1-\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\right)+\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }{\displaystyle \frac{1}{\alpha -1}}\left(\sum _{i=1}^p\left(s\left(a_i\right|b_j)-s{\left(a_i\right|b_j)}^{\alpha }\right)\right)=$$

$$T_{\alpha }\left(B\right)+\sum _{j=1}^{q}s{\left(b_j\right)}^{\alpha }\sum _{i=1}^p{l}_{\alpha }\left(s\left(a_i\right|b_j)\right)\le T_{\alpha }\left(A\right)+T_{\alpha }\left(B\right).$$

□

5. The Conditional Tsallis Entropy

In this section, we introduce the conditional Tsallis entropy of partitions and we give some properties.

 Definition 11. 

If $A,B$ are two partitions, then the conditional Tsallis entropy of A given B is defined by the formula

$$\begin{array}{c}\hfill T_{\alpha }\left(A\right|B)=T_{\alpha }(A\vee B)-T_{\alpha }\left(B\right).\end{array}$$

(1)

Next, lemma is a tool in the proof of Proposition 6.

 Lemma 1. 

Let $A=(a_1,\dots ,a_p),B=(b_1,\dots ,b_q)$ be partitions of unity and $C=A\vee B=c_{ij}$ be the Riesz common refinement of A and B. Then, we have

$$T_{\alpha }\left(A\right|B)={\displaystyle \frac{1}{\alpha -1}}\left(\sum _{j=1}^q{\left(s\left(b_j\right)\right)}^{\alpha }-\sum _{j=1}^q\sum _{i=1}^p{\left(s\left(c_{ij}\right)\right)}^{\alpha }\right).$$

 Proof. 

We get

$$T_{\alpha }\left(A\right|B)=T_{\alpha }(A\vee B)-T_{\alpha }\left(B\right)=$$

$$={\displaystyle \frac{1}{\alpha -1}}(1-\sum _{j=1}^q\sum _{i=1}^p{\left(s\left(c_{ij}\right)\right)}^{\alpha }-1+\sum _{j=1}^q{\left(s\left(b_j\right)\right)}^{\alpha })=$$

$${\displaystyle \frac{1}{\alpha -1}}\left(\sum _{j=1}^q{\left(s\left(b_j\right)\right)}^{\alpha }-\sum _{j=1}^q\sum _{i=1}^p{\left(s\left(c_{ij}\right)\right)}^{\alpha }\right).$$

□

 Proposition 6. 

Let $A,B,C$ be partitions in E. Then

$$T_{\alpha }(A\vee B|C)=T_{\alpha }\left(A\right|C)+T_{\alpha }\left(B\right|A\vee C).$$

 Proof. 

Let $A=(a_1,\dots ,a_n)$, $B=(b_1,\dots ,b_m)$ and $C=(c_1,\dots ,c_p)$. Then, by Lemma 1, we get

$$T_{\alpha }\left(A\right|C)+T_{\alpha }\left(B\right|A\vee C)=$$

$${\displaystyle \frac{1}{\alpha -1}}(\sum _{k=1}^p{\left(s\left(c_k\right)\right)}^{\alpha }-\sum _{i=1}^n\sum _{k=1}^p{\left(s\left(d_{ik}\right)\right)}^{\alpha }+\sum _{i=1}^n\sum _{k=1}^p{\left(s\left(d_{ik}\right)\right)}^{\alpha }-\sum _{j=1}^m\sum _{k=1}^p\sum _{i=1}^n{\left(s\left(e_{ikj}\right)\right)}^{\alpha })=$$

$${\displaystyle \frac{1}{\alpha -1}}(\sum _{k=1}^p{\left(s\left(c_k\right)\right)}^{\alpha }-\sum _{j=1}^m\sum _{k=1}^p\sum _{i=1}^n{\left(s\left(e_{ikj}\right)\right)}^{\alpha })=$$

$$T_{\alpha }(A\vee B|C),$$

where $D=A\vee C=d_{ik}$ is the Riesz common refinement of A and C, and $E=B\vee D=e_{ikj}$ is the Riesz common refinement of D and B. □

6. Tsallis Entropy of a MV-Dynamical System

In this section, we introduce and study $\alpha$-Tsallis entropy of MV-dynamical systems, with $\alpha >1$.

 Definition 12. 

Let $(E,s,\tau )$ be a dynamical system in an MV-algebra E, and let A be a partition of unity in E. Then, we define the strong Tsallis entropy of A by

$$T_{\alpha }^s(A,\tau ):=\underset{n}{lim}{T}_{\alpha }\left({\vee }_{k=0}^{n-1}{\tau }^k\left(A\right)\right).$$

Our next step is the definition of Tsallis entropy of an MV-dynamical system. For that, we need the following propositions:

 Proposition 7. 

Let $(E,s,\tau )$ be a dynamical system in an MV-algebra E, and A be a partition. Then, for any non-negative integer r, the following equality holds:

$$T_{\alpha }^s(A,\tau )=T_{\alpha }^s({\vee }_{i=0}^r{\tau }^i\left(A\right),\tau ).$$

 Proof. 

Using the definition, we can write

$$T_{\alpha }^s({\vee }_{i=0}^r{\tau }^i\left(A\right),\tau )=\underset{n}{lim}{T}_{\alpha }\left({\vee }_{i=0}^{n-1}{\tau }^i\left({\vee }_{i=0}^r{\tau }^i\left(A\right)\right)\right)$$

$$=\underset{n}{lim}{T}_{\alpha }\left({\vee }_{i=0}^{r+n-1}{\tau }^i\left(A\right)\right)=$$

$$\underset{n}{lim}{T}_{\alpha }\left({\vee }_{k=0}^{n-1}{\tau }^k\left(A\right)\right)=T_{\alpha }^s(A,\tau ).$$

□

 Proposition 8. 

Let $A,B$ be partitions such that $A\le B$. Then,

$$T_{\alpha }^s(A,\tau )\le T_{\alpha }^s(B,\tau ).$$

 Proof. 

Suppose $A\le B$.

Now if $P_i,Q_i,(i=1,2)$ are partitions with $P_1\le Q_1$ and $P_2\le Q_2$, then $P_1\vee P_2\le Q_1\vee Q_2$. So we have by induction

$${\vee }_{i=0}^{n-1}{\tau }^i\left(A\right)\le {\vee }_{i=0}^{n-1}{\tau }^i\left(B\right),$$

for every $n\in \mathbb{N}$. Therefore, we get

$$T_{\alpha }\left({\vee }_{i=0}^{n-1}{\tau }^i\left(A\right)\right)\le T_{\alpha }\left({\vee }_{i=0}^{n-1}{\tau }^i\left(B\right)\right).$$

Consequently, we get $T_{\alpha }^s(A,\tau )\le T_{\alpha }^s(B,\tau )$. □

 Definition 13. 

We define the Tsallis entropy of $(E,s,\tau )$ as

$$T_{\alpha }\left(E\right)=sup\left\{T_{\alpha }^s(A,\tau )\right|A\mathit{partition}\mathit{of}E\}.$$

Notice that the Tsallis entropy of an MV-dynamical system $(E,s,\tau )$ does not depend on $\tau$, as the following theorem shows:

 Theorem 2. 

Let $(E,s,\tau )$ be an MV-dynamical system. Then, the following two numbers are equal:

$$T_1=sup\left\{T_{\alpha }\left(A\right)\right|\mathit{A}\mathit{partition}\mathrm{of}\mathit{E}\},$$

$$T_2=T_{\alpha }\left(E\right)=sup\left\{T_{\alpha }(A\vee \dots \vee {\tau }^n\left(A\right))\right|A\mathit{partition}\mathit{of}\mathit{E},n\in \mathbb{N}\}.$$

 Proof. 

Let A be a partition and $n\in \mathbb{N}$.

Then, we have $A\le A\vee \tau \left(A\right)\vee \dots \vee {\tau }^n\left(A\right);$ hence, by Proposition 4,

$$T_{\alpha }\left(A\right)\le T_{\alpha }(A\vee \tau \left(A\right)\vee \dots \vee {\tau }^n\left(A\right)).$$

Therefore, $T_1\le T_2$. Moreover, we get $T_2\le T_1$, since $T_1$ is the sup of a set larger than the one which gives $T_2$; so $T_1=T_2$. □

 Corollary 1. 

The Tsallis entropy of $(E,s,\tau )$ is independent of τ.

 Example 1. 

Let $E=\{0,1/2,1\}$ be the finite MV chain with three elements. Consider the dynamical system $(E,s,\tau )$, where s is the unique state defined on E, $s\left(t\right)=t$ for any t, and τ is the identity map. The only partitions of unity in E (up to zeros) are $P_1:=(1/2,1/2)$ and $P_2:=(0,1)$. Then,

$$T_2\left(E\right)=H_L\left(E\right)={\displaystyle \frac{1}{2}}.$$

Indeed, $T_2^s(P_1,\tau )=1/2(1-1/2)+1/2(1-1/2)=1/4+1/4=1/2$ and $T_2^s(P_2,\tau )=0(1-0)+1(1-1)=0$.

We calculate $T_{\alpha }^s(P_2,\tau )={\displaystyle \frac{1}{\alpha -1}}[(1-(0^{\alpha }+1^{\alpha })]=0$ and

$$T_{\alpha }^s(P_1,\tau )={\displaystyle \frac{1}{\alpha -1}}[1-({{\displaystyle \frac{1}{2}}}^{\alpha }+{{\displaystyle \frac{1}{2}}}^{\alpha })]={\displaystyle \frac{1}{\alpha -1}}(1-{(1/2)}^{\alpha -1}).$$

Hence,

$$T_{\alpha }\left(E\right)={\displaystyle \frac{1}{\alpha -1}}(1-{(1/2)}^{\alpha -1}).$$

 Example 2. 

Let $E=[0,1]$ be the standard MV-algebra. Consider the dynamical system $(E,s,\tau )$, where s is the unique state defined on E, $s\left(t\right)=t$ for any t, and τ is the identity map.

Consider, for every $n\in \mathbb{N}$, the partition $A_n=(1/n,\dots ,1/n)$, where $1/n$ appears n times; we get by Proposition 3

$$T_{\alpha }^s(A_n,\tau )={\displaystyle \frac{1}{\alpha -1}}\left(1-{{\displaystyle \frac{1}{n}}}^{\alpha -1}\right).$$

Hence,

$$T_{\alpha }\left(E\right)={\displaystyle \frac{1}{\alpha -1}},$$

in particular,

$$T_2\left(E\right)=H_L\left(E\right)=1.$$

The main result of this section is Theorem 3.

 Theorem 3. 

Let $(E_1,s_1,{\tau }_1),(E_2,s_2,{\tau }_2)$ be isomorphic MV-dynamical systems. Then, $T_{\alpha }\left(E_1\right)=T_{\alpha }\left(E_2\right).$

 Proof. 

Let a mapping $f:E_1\to E_2$ represent an isomorphism of $(E_1,s_1,{\tau }_1)$ and $(E_2,s_2,{\tau }_2)$. Let $A=(a_1,\dots ,a_p)$ be a partition in $E_1$; then, $f\left(A\right)=(f\left(a_1\right),\dots ,f\left(a_p\right))$ is a partition in $E_2$. We get

$$T_{\alpha }\left(f\left(A\right)\right)=\sum _{i=1}^p{l}_{\alpha }\left(s_2\left(f\left(a_i\right)\right)\right)=\sum _{i=1}^p{l}_{\alpha }\left(s_1\left(a_i\right)\right)=T_{\alpha }\left(A\right),$$

Hence, we derive—since the supremum on the left side of the inequality is taken over all partitions in $E_1$ and the supremum on the right side of the inequality is taken over all partitions in $E_2$—the following:

$$T_{\alpha }\left(E_1\right)\le T_{\alpha }\left(E_2\right).$$

We can argue analogously for obtaining the opposite inequality, so

$$T_{\alpha }\left(E_2\right)\le T_{\alpha }\left(E_1\right);$$

hence, the thesis comes. □

We now give an example which shows that Tsallis entropy allows one to distinguish some non isomorphic dynamical systems. From now on, let $\alpha >1$.

 Definition 14. 

Let us define a partition A of E α-weakly mixing if the sequence ${\left(T_{\alpha }\left(A\vee \tau \left(A\right)\vee \dots \vee {\tau }^n\left(A\right)\right)\right)}_{n\in \mathbb{N}}$ is strictly increasing.

 Remark 8. 

The existence of an α-weakly mixing partition is an invariant of E under isomorphisms. So, weakly mixing partitions can be used to distinguish certain pairs of dynamical systems modulo isomorphism. Note that, unlike Tsallis entropies, which do not depend on τ, the existence of an α-weakly mixing partition in $(E,s,\tau )$ does depend on τ.

Let us consider the following example. It is very similar to the one contained in [1]; nevertheless, for the sake of completeness, we offer details.

 Example 3. 

Let $\theta \in \mathbb{R}$, and consider the dynamical system $S_{\theta }=(E,s,{\tau }_{\theta })$ (the circle) with

• E is the Riesz MV-algebra of Lebesgue measurable functions from S to $[0,1]$, where $S:=\{(x,y)\in {\mathbb{R}}^2|x^2+y^2=1\}$;
• $s={\displaystyle \frac{l}{2\pi }}$, where l is the Lebesgue integral;
• for every $f\in E$, ${\tau }_{\theta }\left(f\right)\left(x\right)=f\left(x{e}^{i\theta }\right)$ (we compose f with a rotation of the angle θ).

 Proposition 9. 

Let $S_{\theta }$ be as in Example 3. Then, it admits an α-weakly mixing partition if and only if $\theta /2\pi$ is irrational.

 Proof. 

Let $\theta /2\pi =m/n\in \mathbb{Q}$. Then, ${\tau }_{\theta }^n={\tau }_{n\theta }={\tau }_{2\pi m}=identity$ and, for every partition A, the sequence ${\left(T_{\alpha }\left(A\vee {\tau }_{\theta }\left(A\right)\vee \dots \vee {\tau }_{\theta }^m\left(A\right)\right)\right)}_{m\in \mathbb{N}}$ is constant for $m\ge n$.

Let $\theta /2\pi \notin \mathbb{Q}$. Take the partition $A=(S^{+},S^{-})$, where $S^{+}$ is the characteristic function of the upper half of S (including the extremes), and $S^{-}$ is the characteristic function of the lower half of S (excluding the extremes).

Let $n\in \mathbb{N}$, and consider $A_n:=A\vee {\tau }_{\theta }\left(A\right)\vee \dots \vee {\tau }_{n\theta }\left(A\right).$Obviously, $A_{n+1}$ refines $A_n$. We claim that $A_{n+1}$ properly refines $A_n$, that is, it has more blocks of nonzero measure.

In fact, $A_n$ consists of finitely many arcs whose extremes form the set

$$\{0,\pi ,\theta ,\theta +\pi ,2\theta ,2\theta +\pi ,\dots ,n\theta ,n\theta +\pi \}.$$

Indeed, for every $i=1,2,\dots ,n,$ the partition ${\tau }_{\theta }^i\left(A\right)$ is given by the two half circles of extremes $i\theta$ and $i\theta +\pi$. Consider the arc ${\tau }_{(n+1)\theta }\left(S^{+}\right)$. One of its extremes is $(n+1)\theta$, and the equation $(n+1)\theta =m\theta +k\pi$, with $m\le n$ and $k\in \mathbb{Z}$, is impossible, as $\theta /2\pi$ is irrational. Hence, the arc ${\tau }_{(n+1)\theta }\left(S^{+}\right)$ divides some element of the partition $A_n$ into two smaller arcs of positive measure.

Therefore, $A_{n+1}$ properly refines $A_n$, and by definition of $T_{\alpha }$, since by Remark 7 $l_{\alpha }$ is strictly concave, we get $T_{\alpha }\left(A_{n+1}\right)>T_{\alpha }\left(A_n\right)$. So the partition A is $\alpha$-weakly mixing in $S_{\theta }$. □

The following corollary follows from the previous proposition:

 Corollary 2. 

Let $S_{\theta }$ be as in Example 3. Let $\theta \in \mathbb{Q}$ and ${\theta }^{\prime }\notin \mathbb{Q}$. Then, $S_{\theta }$ and $S_{{\theta }^{\prime }}$ are not isomorphic.

Another easy proof of the above corollary follows since rational rotations are periodic and irrational rotations are not.

 Remark 9. 

Let $S_{\theta }$ be as in Example 3. Since S admits partitions of measure $(1/n,\dots ,1/n)$ for every n, Tsallis entropy of $S_{\theta }$ is the maximum possible, that is, $1/(\alpha -1)$.

Finally, we conclude the section by providing a further example (the torus) to distinguish MV-algebraic systems which generalizes Example 3.

 Example 4. 

Let $(\theta ,{\theta }^{\prime })\in {\mathbb{R}}^2$, and consider the MV-dynamical system $S_{\theta ,{\theta }^{\prime }}=(E,s,{\tau }_{\theta ,{\theta }^{\prime }})$ with

• E is the Riesz MV-algebra of Lebesgue measurable functions from $S\times S$ to $[0,1]$, where $S:=\{(x,y)\in {\mathbb{R}}^2|x^2+y^2=1\}$;
• $s={\displaystyle \frac{\mu }{4{\pi }^2}}$, where μ is the Lebesgue integral;
• for every $f\in E$, ${\tau }_{\theta ,{\theta }^{\prime }}\left(f\right)(x,y):=f(x{e}^{i\theta },y{e}^{i{\theta }^{\prime }})$ (we compose f with a rotation of x through the angle θ and a rotation of y through the angle ${\theta }^{\prime }$).

It turns out that the system has a weakly mixing partition if and only if at least one of θ and ${\theta }^{\prime }$ is an irrational multiple of π.

 Remark 10. 

Obviously, it is possible generalize Example 3 to n-ary Cartesian powers of S (n-tori).

Finally, notice that, both in Examples 3 and 4, $T_{\alpha }\left(\tau \right)={\displaystyle \frac{1}{\alpha -1}}$ (for $\tau ={\tau }_{\theta }$ and $\tau ={\tau }_{\theta ,{\theta }^{\prime }}$), so it attains its maximum value, since for every n, there is a partition into n pieces of measure ${\displaystyle \frac{1}{n}}$.

7. Conclusions

In this article, we explored Tsallis entropy in MV-algebras.

First, we introduced the concept of Tsallis entropy for partitions in MV-algebras (see Definition 10). In particular, by setting $\alpha =2$, we obtained the logical entropy of partitions in MV-algebras. Then, we presented some properties, e.g., we proved that Tsallis entropy is a submeasure. Finally, we proved that Tsallis entropy is invariant under isomorphisms of MV-dynamical systems and provided two examples showing that Tsallis entropy can distinguish between non-isomorphic MV-dynamical systems.

We hope this paper encourages further research in this field. For instance, it could be interesting to study general MV-dynamical systems $(E,s,\tau )$ where E is the Riesz MV-algebra of Haar-measurable functions from a Lie group G to $[0,1]$, the state s is an integral associated with the Haar measure of G and $\tau$ an operator that preserves the Haar measure (e.g., a translation).