Tsallis Entropy of Product MV-Algebra Dynamical Systems

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α, where α∈(0,1)∪(1,∞), of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order α, where α>1, has the property of sub-additivity. This property allows us to define, for α>1, the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.


Introduction
The Shannon entropy [1] is the foundational concept of information theory (cf. [2]). We remind readers that if an experiment has k outcomes with probabilities p 1 , p 2 , . . . , p k , then its Shannon entropy is defined as the sum ∑ k i=1 s(p i ), where s : [0, 1] → [0, ∞) is Shannon's entropy function defined by: for every x ∈ [0, 1] (0 log 0 is defined to be 0). Many years later, the Shannon entropy was exploited surprisingly in a completely different field, namely, in dynamical systems. Recall that by a dynamical system in the sense of classical probability theory, we understand a system (Ω, Σ, µ, T), where (Ω, Σ, µ) is a probability space, and T : Ω → Ω is a measure µ preserving transformation. The entropy of dynamical systems was introduced by Kolmogorov and Sinai [3,4] as an invariant for distinguishing them. Namely, if two dynamical systems are isomorphic, then they have the same entropy. In this way Kolmogorov and Sinai showed the existence of non-isomorphic Bernoulli shifts. The successful using the Kolmogorov-Sinai entropy of dynamical systems has led to an intensive study of alternative entropy measures of dynamical systems. We note that in Reference [5], the concept of logical entropy H l (T) of a dynamical system (Ω, Σ, µ, T) was introduced and studied. It has been shown that by replacing Shannon's entropy function by the function l : [0, 1] → [0, ∞) defined by: for every x ∈ [0, 1], we obtain the results analogous to the case of Kolmogorov-Sinai entropy theory. The logical entropy H l (T) is invariant under isomorphism of dynamical systems; so it can be used as an alternative instrument for distinguishing them. For some other recently published results regarding the logical entropy measure, we refer, for example, to References [6][7][8][9][10][11][12][13][14][15].
In fact, all of the above mentioned studies are possible in the Kolmogorov probability theory based on the modern integration theory. This allows us to describe and study some of the problems associated with uncertainty. In Reference [16], Zadeh presented another approach to uncertainty when he introduced the concept of a fuzzy set. Whereas the Kolmogorov probability applications are based on objective measurements, the Zadeh fuzzy set theory is based on subjective improvements. One of the first Zadeh papers on the fuzzy set theory was devoted to probability of fuzzy sets (cf. [17]), and therefore, the entropy of fuzzy dynamical systems has also been studied (cf. [18][19][20][21]). We recall that the fuzzy set is a mapping f : Ω → [0, 1], hence, the fuzzy partition of Ω is a family of fuzzy sets A = { f 1 , f 2 , . . . , f k } such that ∑ k i=1 f i = 1. Again we can meet the Shannon formula: [21]). Anyway the most useful tool for describing multivalued processes is an MV-algebra [22], especially after its Mundici's characterization as an interval in a lattice ordered group (cf. [23,24]). At present, this structure is being investigated by many researchers, and it is natural that there are results also regarding the entropy in this structure; we refer, for instance, to References [25,26]. A probability theory was also investigated on MV-algebras; for a review see Reference [27]. Of course, in some probability problems it is needed to introduce a product on an MV-algebra, an operation outside the corresponding group addition. The operation of a product on an MV-algebra was introduced independently by Riečan [28] from the point of view of probability and Montagna [29] from the point of view of mathematical logic. We note that the notion of product MV-algebra generalizes some families of fuzzy sets; an example of product MV-algebra is a full tribe of fuzzy sets (see e.g., [30]).
The appropriate entropy theory of Shannon and Kolmogorov-Sinai type for the product MV-algebras was created in References [31,32]. The logical entropy, the logical divergence and the logical mutual information of partitions in a product MV-algebra were studied in Reference [8]. In the present paper, we extend the study of entropy in product MV-algebras to the case of Tsallis entropy.
The concept of Tsallis entropy was introduced in 1988 by Constantino Tsallis [33] as a base for generalizing the usual statistical mechanics. In its form it is identical with the Havrda-Charvát structural α-entropy [34], introduced in 1967 in the framework of information theory. If P = {p 1 , p 2 , . . . , p k } is a probability distribution, then its Tsallis entropy of order α, where α ∈ (0, 1) ∪ (1, ∞), is defined as the number: The entropic index α describes the deviation of Tsallis entropy from the standard Shannon one. Evidently, if we define, for α ∈ (0, 1) ∪ (1, ∞), the function l α : for every x ∈ [0, 1], then the Formula (3) can be written in the following form: Putting α = 2 in Equation (5), we obtain: which is the logical entropy of a probability distribution P = {p 1 , p 2 , . . . , p k } defined and studied in Reference [6].
In this article we continue studying entropy in a product MV-algebra, by defining and studying the Tsallis entropy of a partition in a product MV-algebra and the Tsallis entropy of product MV-algebra dynamical systems. The rest of the paper is structured as follows. In the following section, preliminaries and related works are given. Our main results are discussed in Sections 3-5. In Section 3, we define and study the Tsallis entropy of a partition in a product MV-algebra. In Section 4, we introduce the concept of conditional Tsallis entropy of partitions in a product MV-algebra and examine its properties. It is shown that the proposed definitions of Tsallis entropy are consistent, in the case of the limit of α going to 1, with the Shannon entropy of partitions studied in Reference [31]. Section 5 is devoted to the study of Tsallis entropy of product MV-algebra dynamical systems. It is proven that the suggested entropy measure is invariant under isomorphism of product MV-algebra dynamical systems. The last section contains a brief summary.

Preliminaries
We begin with recalling the definitions of the basic notions and some known results used in the paper. For defining the notion of MV-algebra, various (but of course equivalent) axiom systems were used (see e.g., [28,50,51]). In this paper, the definition of MV-algebra given by Riečan in Reference [52] is used, which is based on the Mundici representation theorem [23,24]. In view of the Mundici theorem, any MV-algebra may be considered to be an interval of an abelian lattice ordered group. Recall that by an abelian lattice ordered group [53], we mean a triplet (L, +, ≤), where (L, +) is an abelian group, (L, ≤) is a partially ordered set being a lattice and, for every x, y, z ∈ L, x ≤ y =⇒ x + z ≤ y + z.

Definition 1 ([52]
). An MV-algebra is an algebraic structure A = (A, ⊕, * , 0, u) satisfying the following conditions: (i) there exists an abelian lattice ordered group (L, where 0 is the neutral element of (L, +) and u is a strong unit of L (i.e., u ∈ L such that u > 0 and to each x ∈ L there exists a natural number n with the property x ≤ nu); (ii) ⊕ and * are binary operations on A satisfying the following identities: x ⊕ y = (x + y) ∧ u, x * y = (x + y − u) ∨ 0. (ii) if x, y ∈ A such that x + y ≤ u, then s(x + y) = s(x) + s(y).

Definition 3 ([28]).
A product MV-algebra is an algebraic structure (A, ⊕, * , ·, 0, u), where (A, ⊕, * , 0, u) is an MV-algebra and · is an associative and abelian binary operation on A with the following properties: (i) for every x ∈ A, u · x = x; (ii) if x, y, z ∈ A such that x + y ≤ u, then z · x + z · y ≤ u, and z · (x + y) = z · x + z · y.
In the following text, we will briefly write (A, ·) instead of (A, ⊕, * , ·, 0, u). A relevant probability theory for the product MV-algebras was developed by Riečan in Reference [54]; the entropy theory of Shannon and Kolmogorov-Sinai type for the product MV-algebras was proposed in References [31,32]. The logical entropy theory for the product MV-algebras was proposed in Reference [8]. We present the main idea and some results of these theories that will be used in the following text.
It represents an experiment consisting of the realization of X and Y.
is a state on the considered product MV-algebra (A, ·). Evidently, if {E 1 , E 2 , . . . , E k } is a measurable partition of (Ω, Σ, µ), then the k-tuple I E 1 , I E 2 , . . . , I E k is a partition in the product MV-algebra (A, ·).
Example 2. Consider a probability space (Ω, Σ, µ) and the family A of all Σ-measurable functions f : Ω → [0, 1], so called full tribe of fuzzy sets (cf., e.g., [21,30]). The family A is closed under the natural product of fuzzy sets and it is an important case of product MV-algebras. The mapping s : A → [0, 1] defined, for every f ∈ A, by the formula s( f ) = Ω f dµ, is a state on the product MV-algebra (A, ·). The concept of a partition in the product MV-algebra (A, ·) Tcoincides with the notion of a fuzzy partition (cf. [21]).
The definition of entropy of Shannon type of a partition in a product MV-algebra was introduced in [31] as follows.
Definition 4. Let X = (x 1 , x 2 , . . . , x k ) be any partition in a product MV-algebra (A, ·) and s : A → [0, 1] be a state. Then the entropy of X with respect to s is defined by: If X = (x 1 , x 2 , . . . , x k ), and Y = (y 1 , y 2 , . . . , y l ) are two partitions in (A, ·), then the conditional entropy of X given y j ∈ Y is defined by: where: The conditional entropy of X given Y is defined by: It is used the standard convention that 0 log 0 The basis of the logarithm may be any positive real number, but as a rule logarithms to the basis 2 are taken; the entropy is then expressed in bits. If the natural logarithms are taken in the definition, then the entropy is expressed in nats. The entropy of partitions in a product MV-algebra possesses properties corresponding to properties of Shannon's entropy of measurable partitions; more details can be found in Reference [31].
The definition of logical entropy of a partition in a product MV-algebra was introduced in Reference [8] as follows.
Definition 5. Let X = (x 1 , x 2 , . . . , x k ) be a partition in a product MV-algebra (A, ·), and s : A → [0, 1] be a state. Then the logical entropy of X with respect to s is defined by: where l : is the logical entropy function defined by Equation (2). If X = (x 1 , x 2 , . . . , x k ), and Y = (y 1 , y 2 , . . . , y l ) are two partitions in (A, ·), then the conditional logical entropy of X given Y is defined by:

The Tsallis Entropy of Partitions in a Product MV-Algebra
We begin this section with the definition of Tsallis entropy of a partition in a product MV-algebra (A, ·), and then we will examine its properties. In the following, we will suppose that s : A → [0, 1] is a state. Definition 6. Let X = (x 1 , x 2 , . . . , x k ) be a partition in a product MV-algebra (A, ·). Then we define the Tsallis entropy of order α, where α ∈ (0, 1) ∪ (1, ∞), of the partition X with respect to s by: Remark 1. Let us consider the function l α : [0, 1] → [0, ∞) defined by Equation (4). Since ∑ k i=1 s(x i ) = 1, the formula (11) can be expressed in the following form: Evidently, if we put α = 2, the logical entropy H s l (X) is obtained. It is possible to verify that the function l α is, for every α ∈ (0, 1) ∪ (1, ∞), a non-negative function. Namely, if α ∈ (0, 1), then we have Example 3. Let (A, ·) be any product MV-algebra. Let us consider the partition E = (u) representing an experiment resulting in a certain event. Then E ≺ X, for every partition X in (A, ·), and T s . , x k ) be any partition in a product MV-algebra (A, ·). Then: The equality T s α (X) = 1 α−1 1 − k 1−α holds if and only if the state s is uniform over X, i.e., if and only if s(x i ) = 1 k , for i = 1, 2, . . . , k.
Proof. The inequality T s α (X) ≥ 0 follows from the non-negativity of function l α , so it is sufficient to prove the second assertion. We will use the Jensen inequality. It is easy to verify that the function l α is concave, therefore, applying the Jensen inequality, we have: with the equality if and only if s( The equality holds if and only if s( The following propositions will be needed for the proofs of our results. Proposition 1. Let X, Y, Z be partitions in a product MV-algebra (A, ·). Then: Proof. For the proof, see Reference [8].
As an immediate consequence of the previous theorem and Proposition 1, we obtain the following result.
Theorem 3. Let X, Y be partitions in a product MV-algebra (A, ·). Then, for α > 1, it holds: Proof. Suppose that X = (x 1 , x 2 , . . . , x k ), Y = (y 1 , y 2 , . . . , y l ). Let us calculate: The last inequality combined with (13) yields the claim. □ Theorem 3. Let , XY be partitions in a product MV-algebra ( , ). A  Then, for   1, it holds: Evidently, X and Y are partitions in the product MV-algebra (A, ·). By elementary calculations we get that they have the state values 1 2 , 1 2 and 1 3 , 2 3 of the corresponding elements, respectively. The partition X ∨ Y = ( f 1 · g 1 , f 1 · g 2 , f 2 · g 1 , f 2 · g 2 ) has the state values 1 4 , 1 4 , 1 12 , 5 12 of the corresponding elements. We want to find out whether the statement of the previous theorem is true in the case under consideration. Using the formula (11), it can be computed that T s 2 (X) = 0.5, T s 2 (Y) , which is consistent with the assertion of Theorem 3. Put α = 1 2 . We obtain: T s 1/2 (X) It can be seen that T s 1/2 (X ∨ Y) > T s 1/2 (X) + T s 1/2 (Y). The result means that the Tsallis entropy T s α (X) of order α ∈ (0, 1) does not have the property of sub-additivity.
One of the most important properties of Shannon entropy is additivity. In the following theorem it is shown that the Tsallis entropy T s α (X) does not have the property of additivity; it satisfies the following weaker property of pseudo-additivity.
Evidently, X and Y are partitions in the product MV-algebra ( , ).
A    In the last part of this section, we will prove the concavity of Tsallis entropy T s α (X) on the family of all states defined on a given product MV-algebra (A, ·). Proposition 5. Let s 1 , s 2 be two states defined on a common product MV-algebra (A, ·). Then, for every real number λ ∈ [0, 1], the map λs 1 + (1 − λ)s 2 : A → [0, 1] is a state on (A, ·).
Proof. The proof is simple, so it is omitted. Theorem 5. Let s 1 , s 2 be two states defined on a common product MV-algebra (A, ·). Then, for every partition X in a product MV-algebra (A, ·), and for every real number λ ∈ [0, 1], the following inequality holds: Proof. Assume that X = (x 1 , x 2 , . . . , x k ). The function l α is concave, therefore, for every real number λ ∈ [0, 1], we get: As a consequence of Theorem 5, we get the concavity of the logical entropy H s l (X) as a function of s. The result of the previous theorem is illustrated in the following example. . By simple calculation we get that it has the s 1 -state values 1 3 , 2 3 of the corresponding elements, and the s 2 -state values 1 9 , 8 9 of the corresponding elements. In the previous theorem we put λ = 0.2. We will show that, for the chosen α ∈ (0, 1) ∪ (1, ∞), the following inequality holds: Put α = 1 2 . We calculated that T = 0.6267. One can easily check that in this case: For the case of α = 2, i.e., for the logical entropy, we get: T s 1 2 (X)

The Conditional Tsallis Entropy of Partitions in a Product MV-Algebra
In this section we introduce and study the concept of conditional Tsallis entropy of partitions in a product MV-algebra (A, ·). Definition 7. Let X = (x 1 , x 2 , . . . , x k ), and Y = (y 1 , y 2 , . . . , y l ) be partitions in a product MV-algebra (A, ·). We define the conditional Tsallis entropy of order α, where α ∈ (0, 1) ∪ (1, ∞), of X given Y as the number: Remark 2. Evidently, if we put α = 2, then we obtain the conditional logical entropy of X given Y defined by Equation (10).
At α = 1 the value of T s α (X/Y) is undefined because it gives the shape 0 0 . In the following theorem it is shown that for α → 1 the conditional Tsallis entropy T s α (X/Y) tends to the conditional Shannon entropy H s (X/Y) defined by the formula (8), when the natural logarithm is taken in this formula. Theorem 6. Let X = (x 1 , x 2 , . . . , x k ), and Y = (y 1 , y 2 , . . . , y l ) be partitions in a product MV-algebra (A, ·). Then: Proof. For every α ∈ (0, 1) ∪ (1, ∞), we have: where f and g are continuous functions defined, for every α ∈ (0, ∞), by the equalities: The functions f and g are differentiable and evidently, lim α→1 g(α) = 0. Also, it can easily be verified that lim α→1 f (α) = 0. Indeed, by Proposition 3, we get: Using L'Hôpital's rule, it follows that lim , under the assumption that the right hand side exists. It holds d dα g(α) = 1, and: It follows that: s(y j ) .
Example 6. Let X = (x 1 , x 2 , . . . , x k ) be any partition in a product MV-algebra (A, ·), and E = (u). Then: Theorem 7. Let X = (x 1 , x 2 , . . . , x k ) be any partition in a product MV-algebra (A, ·). Then: Proof. The statement is an immediate consequence of the previous theorem; it suffices to put Y = E = (u).
By combining the property (iii) from Theorem 8 with Theorem 4, we obtain the following property of conditional Tsallis entropy T s α (X/Y).

Theorem 9.
If partitions X, Y in a product MV-algebra (A, ·) are statistically independent with respect to s, then: Theorem 10. Let X, Y be partitions in a product MV-algebra (A, ·). Then, for α > 1, it holds: Proof. Let α > 1. Then by the use of the property (iii) from Theorem 8 and Theorem 3, we get: To illustrate the result of previous theorem, we provide the following example, which is a continuation of Example 4. = 0.7877. By easy calculations we get that T s 3 (X/Y) It can be seen that T s 3 (X/Y) < T s 3 (X), and T s 3 (Y/X) < T s 3 (Y), which is consistent with the assertion of Theorem 10. On the other hand, we have T s 1/2 (X/Y) > T s 1/2 (X), and T s 1/2 (Y/X) > T s 1/2 (Y). The result means that the conditional Tsallis entropy T s α (X/Y) of order α ∈ (0, 1) does not have the property of monotonicity.

The Tsallis Entropy of Dynamical Systems in a Product MV-Algebra
In this section, we introduce and study the concept of the Tsallis entropy of a dynamical system in a product MV-algebra (A, ·). Definition 8. ( [32]). By a dynamical system in a product MV-algebra (A, ·), we understand a system (A, s, τ), where s : A → [0, 1] is a state, and τ : A → A is a map such that τ(u) = u, and, for every x, y ∈ A, the following conditions are satisfied:

Remark 3.
We say also briefly a product MV-algebra dynamical system instead of a dynamical system in a product MV-algebra. Example 8. Let (Ω, Σ, µ, T) be a classical dynamical system. Let us consider the product MV-algebra (A, ·) and the state s : A → [0, 1] from Example 1. In addition, let us define the mapping τ : A → A by the equality τ(I E ) = I E • T = I T −1 (E) , for every I E ∈ A. Then the system (A, s, τ) is a dynamical system in the considered product MV-algebra (A, ·). Example 9. Let (Ω, Σ, µ, T) be a classical dynamical system. Let us consider the product MV-algebra (A, ·) and the state s : A → [0, 1] from Example 2. If we define the mapping τ : A → A by the equality τ( f ) = f • T, for every f ∈ A, then it is easy to verify that the system (A, s, τ) is a dynamical system in the considered product MV-algebra (A, ·).
algebraic structure. It has been shown that the proposed concepts are consistent, in the case of the limit of α → 1, with the Shannon entropy expressed in nats, defined and studied in Reference [31]. Moreover, putting α = 2 in the proposed definitions, we obtain the logical entropy of partitions in a product MV-algebra defined and studied in Reference [8]. Section 5 was devoted to the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. From Example 8 it follows that the notion of product MV-algebra dynamical system is a generalization of the concept of classical dynamical system. We have shown that the Tsallis entropy is invariant under isomorphism of product MV-algebra dynamical systems.
In the proofs we used L'Hôpital's rule and the known Jensen inequality. To illustrate the results, we have provided several numerical examples. In Example 2, we have mentioned that the full tribe of fuzzy sets is a special case of product MV-algebras; hence, all the results of this article can be directly applied to this family of fuzzy sets. We remind that a fuzzy subset of a non-empty set Ω is any mapping f : Ω → [0, 1], where the value f (ω) is interpreted as the degree of belonging of element ω of Ω to the fuzzy set f (cf. [16]). In Reference [56], Atanassov has generalized the Zadeh fuzzy set theory by introducing the idea of an intuitionistic fuzzy set (IF-set), a set that has the degree of belonging as well as the degree of non-belonging with each of its elements. From the point of view of application, it should be noted that for a given class F of IF-sets can be created an MV-algebra A such that F can be inserted to A. Also the operation of product on F can be defined by such a way that the corresponding MV-algebra is a product MV-algebra. Therefore, the presented results are also applicable to the case of IF-sets.