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Logical entropy of dynamical systems in product MValgebras and general scheme
Advances in Difference Equations volume 2019, Article number: 9 (2019)
Abstract
The present paper is aimed at studying the entropy of dynamical systems in product MValgebras. First, by using the concept of logical entropy of a partition in a product MValgebra introduced and studied by Markechová et al. (Entropy 20:129, 2018), we define the logical entropy of a dynamical system in the studied algebraic structure. In addition, we introduce a general type of entropy of a product MValgebra dynamical system that includes the logical entropy and the Kolmogorov–Sinai entropy as special cases. It is proved that the proposed entropy measure is invariant under isomorphism of product MValgebra dynamical systems.
1 Introduction
The Shannon entropy [2] is the basic notion of information theory (cf. [3]). If an experiment has n results with probabilities \(p_{1},p_{2},\ldots,p_{n}\), then its entropy is the sum \(\sum_{i = 1}^{n} s(p_{i})\), where \(s: [ 0, 1 ] \to [ 0, \infty )\) is Shannon’s entropy function defined by equation
for every \(x \in [ 0, 1 ]\) (\(0\log 0\) is defined to be 0). Many years later, the Shannon entropy was used surprisingly in a quite different area of theory as well as in practice, i.e., in dynamical systems. Recall that a classical dynamical system is a quaternion \((\varOmega, S, P, T)\), where (\(\varOmega, S, P\)) is a probability space and \(T:\varOmega \to \varOmega\) is a measure preserving map, i.e., \(P(T^{  1}(B)) = P(B), B \in S\). If \(\mathcal{B}= \{ B_{1},B_{2},\ldots,B_{n} \}\) is a measurable partition of Ω with probabilities \(p_{1},p_{2},\ldots,p_{n}\) of the corresponding elements, then its entropy is again \(H(\mathcal{B})= \sum_{i = 1}^{n} s(p_{i}) =  \sum_{i = 1}^{n} p_{i} \cdot \log p_{i}\). If \(\mathcal{B}= \{ B_{1},B_{2},\ldots,B_{n} \}\) and \(\mathcal{C}= \{ C_{1},C_{2},\ldots,C_{m} \}\) are measurable partitions of Ω, then the measurable partition \(\mathcal{B}\vee\mathcal{C} = \{ B_{i} \cap C_{j}; i = 1,2,\ldots,n,j = 1,2,\ldots,m \}\) represents an experiment consisting of a realization of experiments \(\mathcal{B}\) and \(\mathcal{C}\). Further, by \(T^{  1}(\mathcal{B})\) the measurable partition \(\{ T^{  1}(B_{1}),T^{  1}(B_{2}),\ldots,T^{  1}(B_{n}) \}\) is denoted. The entropy of a dynamical system (\(\varOmega, S, P, T\)) has been defined by Kolmogorov and Sinai [4, 5] as the number \(H(T) = \sup H(\mathcal{B}, T)\); \(\mathcal{B}\) is a finite measurable partition of \(\{\varOmega \}\), where \(H(\mathcal{B}, T)= \lim_{n \to \infty} \frac{1}{n}H( \bigvee_{i = 0}^{n  1}T^{  i}(\mathcal{B}))\). It is used to measure dynamical complexity of the considered dynamical system. The number \(H(T)\) is also a useful instrument for distinguishing dynamical systems. Namely, if two dynamical systems are isomorphic, then they have the same entropy. By this way Kolmogorov and Sinai showed that there are nonisomorphic Bernoulli shifts. Recall that the opposite implication holds, but only for Bernoulli shifts: if two Bernoulli shifts have the same entropy, they are isomorphic [6, 7].
The successful using of the Kolmogorov and Sinai entropy of dynamical systems has led to an intensive study of various aspects of alternative entropy measures of dynamical systems. We note that in the recently published paper [8], the notion of logical entropy \(H_{l}(T)\) of a dynamical system (\(\varOmega, S, P, T\)) was proposed and studied. It has been shown that by replacing the Shannon entropy function by the logical entropy function \(l: [ 0, 1 ] \to [ 0, \infty )\) defined by
for every \(x \in [ 0, 1 ]\), we get the results that are analogous to the case of classical Kolmogorov–Sinai entropy theory. It has been proven that the logical entropy \(H_{l}(T)\) distinguishes nonisomorphic dynamical systems; so it can be used as an alternative instrument for distinguishing them. Note that some other recently published results regarding the logical entropy measure can be found, for example, in [9,10,11,12,13,14,15,16,17].
Actually, all of the abovementioned studies are possible in the Kolmogorov probability theory based on the modern integration theory. It gives a possibility to describe and study some problems of uncertainty. Of course, in 1965, Zadeh presented another approach to uncertainty in his article [18]. While the Kolmogorov probability applications are based on objective measurements, the Zadeh fuzzy theory is based on subjective improvements. Of course, one of the first Zadeh articles on the fuzzy set theory was devoted to probability on fuzzy sets (cf. [19]). Therefore, the entropy of fuzzy dynamical systems has also been studied (cf. [20,21,22,23]). Recall that the fuzzy set is a mapping \(f:\varOmega \to [ 0, 1 ]\) (\(f(\omega )\) is interpreted as the degree of the element \(\omega \in \varOmega\) to the considered fuzzy set f), hence the fuzzy partition of Ω is a family of fuzzy sets \(A = \{ f_{1},f_{2},\ldots,f_{n} \}\) such that \(\sum_{i = 1}^{n} f_{i} = 1\). And again we can meet the Shannon formula: \(H(A) =  \sum_{i = 1}^{n} p_{i}\log p_{i}\), where \(p_{i} = \int_{\varOmega} f_{i} \,dP\) (cf. [23]). An overview of publications devoted to the entropy of fuzzy dynamical systems can be found in [24].
In [25], Atanassov presented a remarkable generalization of fuzzy sets, i.e., intuitionistic fuzzy sets. An intuitionistic fuzzy set is a pair \(A = (f_{A}, g_{A})\) of fuzzy sets such that \(f_{A} + g_{A} \le 1\). Here \(f_{A}\) is a membership function, \(g_{A}\) a nonmembership function. If f is a fuzzy set, then the pair (\(f, 1  f\)) is an intuitionistic fuzzy set. Also, the probability on families of intuitionistic fuzzy sets has been studied (cf. [26]).
Anyway, the most useful instrument for describing multivalued processes is an MValgebra [27], especially after its Mundici’s characterization as an interval in a lattice ordered group (cf. [28]). This algebraic structure is currently being studied by many researchers and it is natural that there are many results also regarding entropy in this structure; we refer, for instance, to [29, 30]. A probability theory was studied on MValgebras as well; for a review, see [31]. Of course, in some problems of probability it is necessary to introduce a product on an MValgebra, an operation outside the corresponding group addition. The operation of a product on an MValgebra was introduced independently by Riečan [32] from the point of view of probability and by Montagna [33] from the point of view of mathematical logic. Also, the approach from the point of view of a general algebra proposed by Jakubík in [34] seems to be interesting; see also [35]. We note that the notion of product MValgebra generalizes some families of fuzzy sets; an example of product MValgebra is a full tribe of fuzzy sets (see, e.g., [24]).
A suitable entropy theory of Shannon and Kolmogorov–Sinai type for the product MValgebras has been provided by Petrovičová in [36, 37]. We remark that in our article [38], based on the results of Petrovičová, we introduced the notions of Kullback–Leibler divergence and mutual information of partitions in a product MValgebra. The logical entropy, the logical divergence, and the logical mutual information of partitions in a product MValgebra were studied in [1]. In the present paper, we extend the study of logical entropy of partitions in product MValgebras to the case of product MValgebra dynamical systems. Moreover, we introduce a general type of entropy of a dynamical system in a product MValgebra. The proposed definition is based on the concept of the subadditive generator φ introduced by the authors in [39].
The rest of the article is organized as follows. Section 2 contains basic definitions, notations, and some known facts that will be used in the paper. Our results are presented in the succeeding two sections. In Sect. 3, we define and study the logical entropy of a dynamical system in a product MValgebra and examine its properties. In Sect. 4, a general type of entropy of a dynamical system in a product MValgebra is introduced. It is proved that the proposed entropy measure is invariant under isomorphism of product MValgebra dynamical systems. It is shown that the logical entropy and the Kolmogorov–Sinai entropy of a dynamical system in a product MValgebra can be obtained as special cases of the proposed general scheme. It follows that the isomorphic product MValgebra dynamical systems have the same logical entropy and the same Kolmogorov–Sinai entropy. We illustrate the results with examples. Finally, the last section provides brief closing remarks.
2 Basic definitions and related works
We start by reminding the definitions of basic terms and some of the known results that will be used in the article. We mention some works related to the subject of this article, of course, without claiming completeness.
Several different (but equivalent) axiom systems have been used to define the term of MValgebra (cf., e.g., [32, 40, 41]). In our article, we apply the definition of MValgebra in accordance with the definition given by Riečan in [42], which is based on the Mundici representation theorem. Based on Mundici’s theorem [28] (see also [43]), MValgebras can be viewed as intervals of an abelian latticeordered group (shortly lgroup). We remind that by an lgroup (cf. [44]) we understand a triplet \((G, +, \le )\), where (\(G, + \)) is an abelian group, (\(G, \le \)) is a partially ordered set being a lattice, and \(x \le y \Longrightarrow x + z \le y + z\).
Definition 2.1
([42])
An MValgebra is an algebraic structure \(\mathcal{A}= (A, \oplus, *, 0, u)\) satisfying the following conditions:

(i)
There exists an lgroup (\(G, +, \le \)) such that \(A = [0, u] = \{ x \in G; 0 \le x \le u\}\), where 0 is the neutral element of (\(G, + \)) and u is a strong unit of G (i.e., \(u \in G\) such that \(u > 0\) and to every \(x \in G\) there exists a positive integer n with the property \(x \le nu\));

(ii)
⊕,∗ are binary operations on A satisfying the following identities: \(x \oplus y = (x + y) \wedge u, x * y = (x + y  u) \vee 0\).
Definition 2.2
([31])
A state on an MValgebra \(\mathcal{A}= (A, \oplus, *, 0, u)\) is a mapping \(\mu:A \to [0, 1]\) with the following two properties:

(i)
\(\mu (u) = 1\);

(ii)
If \(x,y \in A\) such that \(x + y \le u\), then \(\mu (x + y) = \mu (x) + \mu (y)\).
Definition 2.3
([42])
A product MValgebra is an algebraic structure \((A, \oplus, *, \cdot, 0, u)\), where (\(A, \oplus, *, 0, u\)) is an MValgebra and ⋅ is an associative and abelian binary operation on A with the following properties:

(i)
For every \(x \in A, u \cdot x = x\);

(ii)
If \(x,y,z \in A\) such that \(x + y \le u\), then \(z \cdot x + z \cdot y \le u\), and \(z \cdot (x + y) = z \cdot x + z \cdot y\).
For brevity, we will write (\(A, \cdot \)) instead of \((A, \oplus, *, \cdot, 0, u)\). A relevant probability theory for the product MValgebras was developed by Riečan in [45], see also [46, 47]; the entropy theory of Shannon and Kolmogorov–Sinai type for the product MValgebras was proposed in [36, 37]. The logical entropy of a partition in a product MValgebra (\(A, \cdot \)) was defined and studied in [1]. We present the main idea and some results of these theories that will be used in the following text.
By a partition in a product MValgebra \((A, \cdot )\), we understand any ntuple \(X = (x_{1},x_{2},\ldots, x_{n})\) of elements of A with the property \(x_{1} + x_{2} +\cdots + x_{n} = u\). In the system of all partitions in a given product MValgebra \((A, \cdot )\), we define the refinement partial order ≻ in a standard way (cf. [1]). If \(X = (x_{1},x_{2},\ldots,x_{n})\) and \(Y = (y_{1},y_{2},\ldots,y_{m})\) are two partitions in \((A, \cdot )\), then we write \(Y \succ X\) (and we say that Y is a refinement of X), if there exists a partition \(\{ I(1),I(2),\ldots,I(n) \}\) of the set \(\{ 1,2,\ldots,m \}\) such that \(x_{i} = \sum_{j \in I ( i )} y_{j}\), for \(i = 1,2,\ldots,n\). Further, we put \(X \vee Y =(x_{i} \cdot y_{j}; i = 1,2,\ldots,n, j = 1,2,\ldots,m)\). Since\(\sum_{i = 1}^{n} \sum_{j = 1}^{m} x_{i} \cdot y_{j} = ( \sum_{i = 1}^{n} x_{i} ) \cdot ( \sum_{j = 1}^{m} y_{j} ) = u \cdot u = u\), the system \(X \vee Y\) is a partition in \((A, \cdot )\); it represents an experiment consisting of a realization of X and Y.
Later we shall need the following assertions:
Proposition 2.1
Let \(X = (x_{1},x_{2},\ldots,x_{n})\) be a partition in a product MValgebra (\(A, \cdot \)) and \(\mu:A \to [0, 1]\) be a state. Then, for any element \(y \in A\), it holds \(\mu (y) = \sum_{i = 1}^{n} \mu (x_{i} \cdot y)\).
Proof
The proof can be found in [1]. □
Proposition 2.2
If \(X, Y,Z\) are partitions in a product MValgebra \((A, \cdot )\), then it holds \(X \vee Y \succ X\), and \(Y \succ X\) implies \(Y \vee Z \succ X \vee Z\).
Proof
The proof can be found in [1]. □
Proposition 2.3
Let \(X, Y,V,Z\) be partitions in a product MValgebra (\(A, \cdot \)) such that \(Y \succ X\) and \(Z \succ V\). Then \(Y \vee Z \succ X \vee V\).
Proof
Assume that \(X = (x_{1},x_{2},\ldots,x_{n}), Y = (y_{1},y_{2},\ldots,y_{m}), V = (v_{1},v_{2},\ldots,v_{p}), Z = (z_{1},z_{2},\ldots, z_{q}), Y \succ X, Z \succ V\). Then there exists a partition \(\{ I(1),I(2),\ldots,I(n) \}\) of the set \(\{ 1,2,\ldots,m \}\) such that \(x_{i} = \sum_{j \in I ( i )} y_{j}\) for \(i = 1,2,\ldots,n\), and there exists a partition \(\{ J(1),J(2),\ldots,J(p) \}\) of the set \(\{ 1,2,\ldots,q \}\) such that \(v_{r} = \sum_{k \in J ( r )} z_{k}\) for \(r = 1,2,\ldots,p\). Put \(I(i,r) = \{ (j,k); j \in I(i), k \in J(r) \}\) for \(i = 1,2,\ldots,n, r = 1,2,\ldots,p\). We get
for \(i = 1,2,\ldots,n, r = 1,2,\ldots,p\), which means that \(Y \vee Z \succ X \vee V\). □
Definition 2.4
Let μ be a state on a product MValgebra \((A, \cdot )\). We say that partitions \(X, Y\) in (\(A, \cdot \)) are statistically independent with respect to μ if \(\mu (x \cdot y) = \mu (x ) \cdot \mu (y)\) for every \(x \in X\) and \(y \in Y\).
The following definition of entropy of Shannon type was introduced in [36].
Definition 2.5
Let \(X = (x_{1},x_{2},\ldots,x_{n})\) be a partition in a product MValgebra \((A, \cdot )\), and \(\mu:A \to [0, 1]\) be a state. Then the entropy of X with respect to μ is defined by Shannon’s formula:
where \(s: [ 0, 1 ] \to [ 0, \infty )\) is the Shannon entropy function defined by Eq. (1.1). If \(X = (x_{1},x_{2},\ldots,x_{n})\) and \(Y = (y_{1},y_{2},\ldots,y_{m})\) are two partitions in \((A, \cdot )\), then the conditional entropy of X given Y is defined by
In Eq. (2.2), it is assumed that \(0 \cdot \log \frac{0}{x} = 0\) if \(x \ge 0\). The entropy and the conditional entropy of partitions in a product MValgebra satisfy all properties corresponding to the properties of Shannon’s entropy of measurable partitions in the classical case; for more details, see [36]. In particular, it holds \(H_{s}^{\mu} ( X \vee Y ) \le H_{s}^{\mu} ( X ) + H_{s}^{\mu} ( Y )\) for every partition \(X, Y\) in \((A, \cdot )\). The equality holds if and only if \(X, Y\) are statistically independent partitions with respect to μ. This means that Shannon’s entropy of partitions in a product MValgebra has the property of additivity and also the property of subadditivity.
The definition of logical entropy of a partition in a product MValgebra was introduced in [1] as follows.
Definition 2.6
Let \(X = (x_{1},x_{2},\ldots,x_{n})\) be a partition in a product MValgebra \((A, \cdot )\), and \(\mu:A \to [0, 1]\) be a state. Then the logical entropy of X with respect to μ is defined by
where \(l: [ 0, 1 ] \to [ 0, \infty )\) is the logical entropy function defined by Eq. (1.2). If \(X = (x_{1},x_{2},\ldots,x_{n})\) and \(Y = (y_{1},y_{2},\ldots,y_{m})\) are two partitions in \((A, \cdot )\), then the conditional logical entropy of X given Y is defined by
The basic properties of the logical entropy of partitions in a product MValgebra were derived in [1]. Specifically, this entropy measure has been shown to have the property of subadditivity, but it does not have the property of additivity. It satisfies the following property: if \(X, Y\) are statistically independent partitions in a product MValgebra \((A, \cdot )\), then:
Moreover, the proposed logical entropy measure has the following properties: (L1) for every partition \(X, Y\) in \((A, \cdot )\), it holds \(H_{{l}}^{\mu} (X \vee Y) = H_{{l}}^{\mu} (X) + H_{{l}}^{\mu} (Y/X)\); (L2) for every partition \(X, Y\) in (\(A, \cdot \)) such that \(Y \succ X\), it holds \(H_{{l}}^{\mu} (Y) \ge H_{{l}}^{\mu} (X)\).
3 The logical entropy of dynamical systems in product MValgebras
In this section, we extend the definition of logical entropy of a partition in a product MValgebra to the case of dynamical systems and prove basic properties of this measure of information. The known Kolmogorov–Sinai theorem on generators is a useful instrument to compute the entropy of a dynamical system. In the final part of this section we provide a logical version of this theorem for the studied case of product MValgebra.
Definition 3.1
([37])
By a dynamical system in a product MValgebra \((A, \cdot )\), we understand a system \((A, \mu, U )\), where \(\mu:A \to [0, 1]\) is a state, and \(U:A \to A\) is a map such that \(U (u) = u\),and, for every \(x,y \in A\), the following conditions are satisfied:

(i)
if \(x + y \le u\), then \(U(x) + U(y) \le u\) and \(U(x + y) = U(x) + U(y)\);

(ii)
\(U(x \cdot y) = U(x) \cdot U(y)\);

(iii)
\(\mu (U (x)) = \mu (x)\).
Remark 3.1
For the sake of brevity, we say also a product MValgebra dynamical system instead of a dynamical system in a product MValgebra.
Example 3.1
Let (\(\varOmega, S, P, T\)) be a classical dynamical system. Put \(A = \{ \chi_{B}; B \in S \}\), where \(\chi_{B}:\varOmega \to \{ 0, 1 \}\) is the characteristic function of the set \(B \in S\). The family A is closed under the product of characteristic functions, and it is a special case of product MValgebras. If we define the mapping \(\mu:A \to [0, 1]\) by \(\mu (\chi_{B}) = P(B), B \in S\), then μ is a state on the product MValgebra \((A, \cdot )\). In addition, let us define the mapping \(U:A \to A\) by the equality \(U(\chi_{B}) = \chi_{B} \circ T = \chi_{T^{  1}(B)}, \chi_{B} \in A\). Then the system (\(A, \mu, U \)) is a dynamical system in the considered product MValgebra \((A, \cdot )\). A measurable partition \(\mathcal{B}= \{ B_{1},B_{2},\ldots,B_{n} \}\) of Ω can be considered as a partition in the product MValgebra \((A, \cdot )\); it suffices to consider \(\chi_{B_{i}}\) instead of \(B_{i}\).
Example 3.2
Let (\(\varOmega, S, P, T\)) be a classical dynamical system. Let A be a family of all Smeasurable functions \(f:\varOmega \to [0, 1]\),the socalled full tribe of fuzzy sets (cf. [24]). The family A is closed also with respect to the natural product of fuzzy sets, and it is an important case of product MValgebras. If we define the state \(\mu:A \to [0, 1]\) by the equality \(\mu (f) = \int_{\varOmega} f \,dP\) for any element f of A, and the mapping \(U:A \to A\) by the equality \(U(f) = f \circ T, f \in A\), then it is easy to verify that the system (\(A, \mu, U \)) is a dynamical system in the considered product MValgebra \((A, \cdot )\). The notion of a partition in the product MValgebra (\(A, \cdot \)) coincides with the notion of a fuzzy partition.
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and \(X = (x_{1},x_{2},\ldots,x_{n})\) be a partition in \((A, \cdot )\). Put \(U(X) = (U(x_{1}),U(x_{2}),\ldots,U(x_{n}))\). Since \(x_{1} + x_{2} +\cdots + x_{n} = u\), according to Definition 3.1, we have \(U(x_{1}) + U(x_{2}) +\cdots + U(x_{n}) = U(x_{1} + x_{2} +\cdots + x_{n}) = U(u) = u\), which means that \(U(X)\) is also a partition in \((A, \cdot )\).
Proposition 3.1
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and \(X, Y\) be partitions in \((A, \cdot )\). Then

(i)
\(U(X \vee Y) = U(X) \vee U(Y)\);

(ii)
\(Y \succ X\) implies \(U(Y) \succ U(X)\).
Proof
(i) Suppose that \(X = (x_{1},x_{2},\ldots,x_{n}), Y = (y_{1},y_{2},\ldots,y_{m})\). By condition (ii) from Definition 3.1, we have
(ii) Suppose that \(X = (x_{1},x_{2},\ldots,x_{n}), Y = (y_{1},y_{2},\ldots,y_{m}), Y \succ X\). Then there exists a partition \(\{ I(1),I(2),\ldots,I(n) \}\) of the set \(\{ 1,2,\ldots,m \}\) such that \(x_{i} = \sum_{j \in I ( i )} y_{j}\) for \(i = 1,2,\ldots,n\). Therefore, by condition (i) from Definition 3.1, we have
However, this means that \(U(Y) \succ U(X)\). □
Define \(U^{2} = U \circ U\), and put \(U^{k} = U \circ U^{k  1}\) for \(k = 1,2,\ldots\) , where \(U^{0}\) is the identical mapping. It is obvious that the map \(U^{k}:A \to A\) satisfies the conditions from Definition 3.1. Hence, for any nonnegative integer k, the system (\(A, \mu, U^{k} \)) is a dynamical system in a product MValgebra \((A, \cdot )\).
Theorem 3.1
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and \(X, Y\) be partitions in \((A, \cdot )\). Then, for any nonnegative integer k, the following equalities hold:

(i)
\(H_{{l}}^{\mu} (U^{k}(X)) = H_{{l}}^{\mu} (X)\);

(ii)
\(H_{{l}}^{\mu} (U^{k}(X)/U^{k}(Y)) = H_{{l}}^{\mu} (X/Y)\).
Proof
Suppose that \(X = (x_{1},x_{2},\ldots,x_{n})\) and \(Y = (y_{1},y_{2},\ldots,y_{m})\).
(i) Since for any nonnegative integer k and \(i = 1,2,\ldots,n\), it holds \(\mu (U^{k}(x_{i})) = \mu (x_{i})\), we obtain
(ii) Based on the same argument, we get
□
Theorem 3.2
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and X be a partition in \((A, \cdot )\). Then, for \(n = 2,3,\ldots\) , the following equality holds:
Proof
We use proof by mathematical induction on n, starting with \(n = 2\). For \(n = 2\), the statement holds by property (L1). We suppose that the statement holds for a given integer \(n > 1\), and we will prove that it is true for \(n + 1\). By property (i) from the previous theorem, we get
Therefore, using (L1) and our inductive hypothesis, we get
In conclusion, the statement holds by the principle of mathematical induction. □
In the following, we will define the logical entropy of a dynamical system \((A, \mu, U )\). First, we define the logical entropy of U relative to a partition X in \((A, \cdot )\). Then we remove the dependence on X to get the logical entropy of a dynamical system \((A, \mu, U )\). We will need the following proposition.
Proposition 3.2
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\). Then, for any partition \(X in (A, \cdot )\), there exists the following limit:
Proof
Put \(c_{n} = H_{{l}}^{\mu} ( \bigvee_{k = 0}^{n  1}U^{k}(X))\) for \(n = 1,2,\ldots\) . Then the sequence \(\{ c_{n} \}_{n = 1}^{\infty} \) is a sequence of nonnegative real numbers satisfying the condition \(c_{r + s} \le c_{r} + c_{s}\) for every \(r, s \in \mathbb{N}\). Indeed, by means of subadditivity of logical entropy and property (i) from Theorem 3.1, we can write
This property guarantees (in view of Theorem 4.9, [48]) the existence of \(\lim_{n \to \infty} \frac{1}{n}c_{n}\). □
Definition 3.2
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and X be a partition in \((A, \cdot )\). Then we define the logical entropy of Urelative to X by
Remark 3.2
Consider any dynamical system (\(A, \mu, U \)) in a product MValgebra \((A, \cdot )\). If we put \(E = (u)\), then E is a partition in (\(A, \cdot \)) such that \(X \succ E\) for any partition X in \((A, \cdot )\), and with the logical entropy \(H_{l}^{\mu} (E) = 0\). Evidently, \(\bigvee_{k = 0}^{n  1}U^{k}(E) = E\), hence \(H_{{l}}^{\mu} (U, E) = 0\).
Theorem 3.3
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and X be a partition in \((A, \cdot )\). Then, for any nonnegative integer r, the following equality holds:
Proof
Using Definition 3.2, we can write
□
Theorem 3.4
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and \(X, Y\) be partitions in (\(A, \cdot \)) such that \(Y \succ X\). Then \(H_{{l}}^{\mu} (U, X) \le H_{{l}}^{\mu} (U, Y)\).
Proof
Let \(Y \succ X\). By Propositions 2.3 and 3.1, we have \(\bigvee_{k = 0}^{n  1}U^{k}(Y) \succ \bigvee_{k = 0}^{n  1}U^{k}(X)\) for \(n = 1,2,\ldots\) . Therefore, by property (L2), we get
Consequently, dividing by n and letting \(n \to \infty\), we get \(H_{{l}}^{\mu} (U, X) \le H_{{l}}^{\mu} (U, Y)\). □
Definition 3.3
The logical entropy of a dynamical system (\(A, \mu, U \)) in a product MValgebra (\(A, \cdot \)) is defined by
Theorem 3.5
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\). Then, for every natural number k, it holds \(H_{{l}}^{\mu} (U^{k}) = k \cdot H_{{l}}^{\mu} (U)\).
Proof
Let X be a partition in \((A, \cdot )\). Then, for every natural number k, we have
Hence we obtain
On the other hand, by Proposition 2.2, we have \(\bigvee_{j = 0}^{k  1}U^{j}(X) \succ X\), and therefore, by Theorem 3.4, we get
It follows from this that
□
In the rest of this section, we formulate a version of the Kolmogorov–Sinai theorem on generators for the case of the logical entropy of a dynamical system \((A, \mu, U )\).
Definition 3.4
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\). A partition Z in (\(A, \cdot \)) is said to be a generator of a dynamical system (\(A, \mu, U \)) if to any partition X in (\(A, \cdot \)) there exists an integer \(k > 0\) such that \(\bigvee_{i = 0}^{k}U^{i}(Z) \succ X\).
Theorem 3.6
Let Z be a generator of a dynamical system \((A, \mu, U )\). Then \(H_{{l}}^{\mu} (U) = H_{{l}}^{\mu} (U, Z)\).
Proof
Let Z be a generator of a dynamical system \((A, \mu, U )\), and X be any partition in \((A, \cdot )\). Then there exists an integer \(k > 0\) such that \(\bigvee_{i = 0}^{k}U^{i}(Z) \succ X\). Therefore, by Theorems 3.4 and 3.3, we have
and consequently
□
4 General type of entropy of dynamical systems in product MValgebras
In this section, we introduce, based on the function \(\varphi:[0, 1] \to \mathbb{R}\), a general type of entropy of a partition in a product MValgebra (\(A, \cdot \)) that contains the Shannon entropy and the logical entropy of a partition in a product MValgebra (\(A, \cdot \)) as special cases. Subsequently, using the concept of φentropy of a partition in \((A, \cdot )\), where φ is a socalled subadditive generator [39], we define a general type of entropy of a dynamical system \((A, \mu, U )\), socalled φentropy of a dynamical system \((A, \mu, U )\). We construct for the proposed entropy measure an isomorphism theory of the Kolmogorov–Sinai type.
Definition 4.1
Let \(X = (x_{1},x_{2},\ldots,x_{n})\) be a partition in a product MValgebra \((A, \cdot )\), and \(\mu:A \to [0, 1]\) be a state. If \(\varphi:[0, 1] \to \mathbb{R}\) is a function, then we define the φ−entropy of X with respect to μ as the number
Example 4.1
If we put \(\varphi = s\), where \(s: [ 0, 1 ] \to [ 0, \infty )\) is the Shannon entropy function defined by Eq. (1.1), then we obtain the Shannon entropy of X, and putting \(\varphi = l\), where \(l: [ 0, 1 ] \to [ 0, \infty )\) is the logical entropy function defined by Eq. (1.2), the logical entropy of X is obtained.
Definition 4.2
([39])
A function \(\varphi: [ 0, 1 ] \to [ 0, \infty )\) is said to be a subadditive generator if the following condition is satisfied: if \(c_{ij} \in [ 0, 1 ], i = 1,2,\ldots,n, j = 1,2,\ldots,m, \sum_{j = 1}^{m} c_{ij} = a_{i}, i = 1,2,\ldots,n, \sum_{i = 1}^{n} c_{ij} = b_{j}, j = 1,2,\ldots,m\), and \(\sum_{i = 1}^{n} a_{i} = 1, \sum_{j = 1}^{m} b_{j} = 1\), then
Remark 4.1
In [39] we have shown that the Shannon entropy as well as the logical entropy functions are subadditive generators. Moreover, a subadditive generator different from these entropy functions was found; it was proven that the function \(k: [ 0, 1 ] \to [ 0, \infty )\) defined by
for every \(x \in [ 0, 1 ]\), is a subadditive generator. The function k will be called the quadratic logical entropy function.
Remark 4.2
Consider any product MValgebra (\(A, \cdot \)) and the partition \(E = (u)\) in \((A, \cdot )\). If \(\varphi:[0, 1] \to \mathbb{R}\) is a function with the property that \(\varphi (1) = 0\) (it is evident that all of the above three entropy functions satisfy this condition), then \(H_{\varphi}^{\mu} (E) = 0\).
Theorem 4.1
Let μ be a state on a product MValgebra \((A, \cdot )\), and φ be a subadditive generator. Then, for any partitions \(X, Y\) in a product MValgebra \((A, \cdot )\), the following inequality holds:
Proof
Suppose that \(X = (x_{1},x_{2},\ldots,x_{n})\) and \(Y = (y_{1},y_{2},\ldots,y_{m})\). Put \(c_{ij} = \mu (x_{i} \cdot y_{j}), a_{i} = \mu (x_{i}), b_{j} = \mu (y_{j})\) for \(i = 1,2,\ldots,n, j = 1,2,\ldots,m\). By Proposition 2.1, we get
for \(i = 1,2,\ldots,n, j = 1,2,\ldots,m\). Further, according to Definition 2.2 and the definition of a partition in a product MValgebra, we have
analogously, we get that \(\sum_{j = 1}^{m} b_{j} = 1\). Hence
□
To illustrate the result of the previous theorem, we provide the following example.
Example 4.2
Consider the measurable space \(([0, 1], B )\), where B is the σalgebra of all Borel subsets of the unit interval \([0, 1]\). Let A be a family of all Borel measurable functions \(f: [0, 1] \to [0, 1]\). If we define in the family A the operation⋅ as the natural product of fuzzy sets, then the system (\(A, \cdot \)) is a product MValgebra. We define a state \(\mu:A \to [0, 1]\) by the equality \(\mu (f) = \int_{0}^{1} f(x)\,dx\) for any element f of A. It is easy to see that the pairs \(X = ( f_{1}, f_{2} ), Y = ( g_{1}, g_{2} )\), where \(f_{1}(x) = x, f_{2}(x) = 1  x, g_{1}(x) = x^{2}, g_{2}(x) = 1  x^{2}, x \in [0, 1]\), are two partitions in (\(A, \cdot \)) with the state values \(\frac{1}{2}, \frac{1}{2}\) and \(\frac{1}{3}, \frac{2}{3}\) of the corresponding elements, respectively. The join of partitions X and Y is the system \(X \vee Y = ( f_{1} \cdot g_{1}, f_{1} \cdot g_{2}, f_{2} \cdot g_{1}, f_{2} \cdot g_{2} )\) with the state values \(\frac{1}{4}, \frac{1}{4},\frac{1}{12},\frac{5}{12}\) of the corresponding elements. By simple calculations we get the Shannon entropies \(H_{s}^{\mu} (X) = 1, H_{s}^{\mu} (Y)\mathbin{\dot{ =} }0.9183, H_{s}^{\mu} (X \vee Y)\mathbin{\dot{ =}} 1.8250\); the logical entropies \(H_{l}^{\mu} (X) = 0.5, H_{l}^{\mu} (Y)\mathbin{\dot{ =}} 0.4444, H_{l}^{\mu} (X \vee Y)\mathbin{\dot{ =}} 0.6944\); and the quadratic logical entropies \(H_{k}^{\mu} (X) = 0.75, H_{k}^{\mu} (Y)\mathbin{\dot{ =}} 0.6666, H_{k}^{\mu} (X \vee Y)\mathbin{\dot{ =}} 0.6615\). It is easy to see that for the subadditive generators \(\varphi = s, \varphi = l\), and \(\varphi = k\), it holds \(H_{\varphi}^{\mu} ( X \vee Y ) \le H_{\varphi}^{\mu} ( X ) + H_{\varphi}^{\mu} ( Y )\), which is consistent with the claim of the previous theorem.
Theorem 4.2
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and \(\varphi:[0, 1] \to \mathbb{R}\) be a function. Then, for any partition X in (\(A, \cdot \)) and for any nonnegative integer k, it holds
Proof
The statement follows immediately from condition (iii) of Definition 3.1. □
Proposition 4.1
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and φ be a subadditive generator. Then, for any partition X in \((A, \cdot )\), there exists the following limit:
Proof
In view of subadditivity of φentropy (Theorem 4.1) and the previous theorem, the proof can be made similarly as the proof of Proposition 3.2. □
Definition 4.3
Let (\(A, \mu, U \)) be a dynamical system in a product MValgebra \((A, \cdot )\), and φ be a subadditive generator. Then we define the φentropy of (\(A, \mu, U \)) by the formula
where
Example 4.3
It is clear that putting \(\varphi = l\), where \(l: [ 0, 1 ] \to [ 0, \infty )\) is the logical entropy function defined by Eq. (1.2), we obtain the logical entropy of a dynamical system \((A, \mu, U )\). If we put \(\varphi = s\), where \(s: [ 0, 1 ] \to [ 0, \infty )\) is the Shannon entropy function defined by Eq. (1.1), we obtain the Kolmogorov–Sinai entropy of a dynamical system (\(A, \mu, U \)) defined and studied by Petrovičová in [37].
Definition 4.4
Two product MValgebra dynamical systems \((A_{1}, \mu_{1}, U_{1} )\), (\(A_{2}, \mu_{2}, U_{2} \)) are said to be isomorphic if there exists some onetoone and onto map \(\psi:A_{1} \to A_{2}\) such that \(\psi (u_{1}) = u_{2}\), and, for every \(x,y \in A_{1}\), the following conditions are satisfied:

(i)
\(\psi (x \cdot y) = \psi (x) \cdot \psi (y)\);

(ii)
if \(x + y \le u_{1}\), then \(\psi (x + y) = \psi (x) + \psi (y)\);

(iii)
\(\mu_{2}(\psi (x)) = \mu_{1}(x)\);

(iv)
\(\psi (U_{1}(x)) = U_{2}(\psi (x))\).
In this case, ψ is called an isomorphism, and we write \(U_{1} \cong U_{2}\).
Proposition 4.2
Let \((A_{1}, \mu_{1}, U_{1} )\),(\(A_{2}, \mu_{2}, U_{2} \)) be isomorphic product MValgebra dynamical systems, and \(\psi:A_{1} \to A_{2}\) be an isomorphism between \((A_{1}, \mu_{1}, U_{1} ), (A_{2}, \mu_{2}, U_{2} )\). Then, for the inverse \(\psi^{  1}:A_{2} \to A_{1}\), the following properties are satisfied:

(i)
\(\psi^{  1}(x \cdot y) = \psi^{  1}(x) \cdot \psi^{  1}(y)\) for every \(x,y \in A_{2}\);

(ii)
if \(x,y \in A_{2}\) such that \(x + y \le u_{2}\), then \(\psi^{  1}(x + y) = \psi^{  1}(x) + \psi^{  1}(y)\);

(iii)
\(\mu_{1}(\psi^{  1} (x)) = \mu_{2}(x)\) for every \(x \in A_{2}\);

(iv)
\(\psi^{  1}(U_{2}(x)) = U_{1}(\psi^{  1} (x))\) for every \(x \in A_{2}\).
Proof
Since \(\psi:A_{1} \to A_{2}\) is bijective, for every \(x,y \in A_{2}\), there exist \(x',y' \in A_{1}\) such that\(\psi^{  1}(x) = x'\) and \(\psi^{  1}(y) = y'\).

(i)
Let \(x,y \in A_{2}\). Then we have
$$\psi^{  1}(x \cdot y) = \psi^{  1}\bigl(\psi \bigl(x'\bigr) \cdot \psi \bigl(y'\bigr)\bigr) = \psi^{  1}\bigl(\psi \bigl(x' \cdot y'\bigr) \bigr) = x' \cdot y' = \psi^{  1}(x) \cdot \psi^{  1}(y). $$ 
(ii)
Let \(x,y \in A_{2}\) such that \(x + y \le u_{2}\). Then \(x' + y' \le u_{1}\), and
$$\psi^{  1}(x + y) = \psi^{  1}\bigl(\psi \bigl(x' \bigr) + \psi \bigl(y'\bigr)\bigr) = \psi^{  1}\bigl(\psi \bigl(x' + y'\bigr)\bigr) = x' + y' = \psi^{  1}(x) + \psi^{  1}(y). $$ 
(iii)
Let \(x \in A_{2}\). Then \(\mu_{2}(x) = \mu_{2}(\psi (x')) = \mu_{1}(x') = \mu_{1}(\psi^{  1} (x))\).

(iv)
Let \(x \in A_{2}\). Then
$$\psi^{  1}\bigl(U_{2}(x)\bigr) = \psi^{  1} \bigl(U_{2}\bigl(\psi \bigl(x'\bigr)\bigr)\bigr) = \psi^{  1}\bigl(\psi \bigl(U_{1}\bigl(x'\bigr) \bigr)\bigr) = U_{1}\bigl(x'\bigr) = U_{1} \bigl(\psi^{  1}(x)\bigr). $$
□
Theorem 4.3
Let φ be a subadditive generator, and \((A_{1}, \mu_{1}, U_{1} )\),(\(A_{2}, \mu_{2}, U_{2} \)) be product MValgebra dynamical systems. If \(U_{1} \cong U_{2}\), then
Proof
Let \(\psi:A_{1} \to A_{2}\) be an isomorphism between dynamical systems \((A_{1}, \mu_{1}, U_{1} ), (A_{2}, \mu_{2}, U_{2} )\). Consider a partition \(X = (x_{1},x_{2},\ldots,x_{n})\) in a product MValgebra \((A_{1}, \cdot )\). Then \(x_{1} + x_{2} +\cdots + x_{n} = u_{1}\), and therefore, by condition (i) of Definition 4.4, it holds \(\psi (x_{1}) + \psi (x_{2}) +\cdots + \psi (x_{n}) = \psi (x_{1} + x_{2} +\cdots + x_{n}) = \psi (u_{1}) = u_{2}\). This means that the collection \(\psi (X) = (\psi (x_{1}),\psi (x_{2}),\ldots,\psi (x_{n}))\) is a partition in a product MValgebra \((A_{2}, \cdot )\). Moreover, according to condition (iii) of Definition 4.4, we have
Hence, using conditions (iv) and (i) of Definition 4.4, we get
Therefore, dividing by n and letting \(n \to \infty\), we obtain
This implies that
and consequently
The converse \(H_{\varphi}^{\mu_{2}}(U_{2}) \le H_{\varphi}^{\mu_{1}}(U_{1})\) is obtained in a similar way; according to Proposition 4.2, it suffices to consider the inverse \(\psi^{  1}:A_{2} \to A_{1}\). □
By combining the previous results, we obtain the following statement.
Corollary 4.1
If \(U_{1} \cong U_{2}\), then

(i)
\(H_{s}^{\mu_{1}}(U_{1}) = H_{s}^{\mu_{2}}(U_{2})\);

(ii)
\(H_{l}^{\mu_{1}}(U_{1}) = H_{l}^{\mu_{2}}(U_{2})\);

(iii)
\(H_{k}^{\mu_{1}}(U_{1}) = H_{k}^{\mu_{2}}(U_{2})\).
Remark 4.3
It trivially follows from the above theorem that if \(H_{\varphi}^{\mu_{1}}(U_{1}) \ne H_{\varphi}^{\mu_{2}}(U_{2})\), then the corresponding dynamical systems \((A_{1}, \mu_{1}, U_{1} )\),(\(A_{2}, \mu_{2}, U_{2} \)) are not isomorphic. This means that the proposed φentropy distinguishes nonisomorphic product MValgebra dynamical systems.
5 Conclusions
In the paper we have extended the results concerning the logical entropy of partitions in product MValgebras provided in [1] to the case of dynamical systems. By using the concept of logical entropy of a partition in a product MValgebra, we introduced the notion of logical entropy of a product MValgebra dynamical system and derived the basic properties of this measure of information. In particular, a logical version of the Kolmogorov–Sinai theorem on generators was provided.
In addition, using the concept of the subadditive generator φ introduced by the authors in [39], we have defined a general type of entropy of a product MValgebra dynamical system \((A, \mu, U )\), the socalled φentropy of a dynamical system \((A, \mu, U )\). The proposed φentropy includes the logical entropy and the Kolmogorov–Sinai entropy as special cases: if we put \(\varphi = l\), where \(l: [ 0, 1 ] \to [ 0, \infty )\) is the logical entropy function defined by Eq. (1.2), we obtain the logical entropy of a dynamical system \((A, \mu, U )\), and putting \(\varphi = s\), where \(s: [ 0, 1 ] \to [ 0, \infty )\) is the Shannon entropy function defined by Eq. (1.1), we obtain the Kolmogorov–Sinai entropy of a dynamical system (\(A, \mu, U \)) defined and studied by Petrovičová in [37]. For the proposed φentropy \(H_{\varphi}^{\mu} (U)\), we have created an isomorphism theory of the Kolmogorov–Sinai type. It was shown that the φentropy \(H_{\varphi}^{\mu} (U)\) distinguishes nonisomorphic product MValgebra dynamical systems. In this way, we have acquired several instruments to distinguish nonisomorphic product MValgebra dynamical systems: the logical, the Kolmogorov–Sinai, and the quadratic logical entropy of a dynamical system \((A, \mu, U )\).
As mentioned above (see Example 3.2), the full tribe of fuzzy sets represents a special case of product MValgebras; the obtained results can therefore be immediately applied to this significant family of fuzzy sets. From the point of view of applications, it is interesting that to a given family \(\mathcal{F}\) of intuitionistic fuzzy sets can be constructed an MValgebra \(\mathcal{A}\) such that \(\mathcal{F}\) can be embedded to \(\mathcal{A}\). Also, product on \(\mathcal{F}\) can be introduced by such a way that the corresponding MValgebra is an MValgebra with product. Hence all results of our paper can be applied also to the case of intuitionistic fuzzy sets.
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Markechová, D., Riečan, B. Logical entropy of dynamical systems in product MValgebras and general scheme. Adv Differ Equ 2019, 9 (2019). https://doi.org/10.1186/s1366201919462
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DOI: https://doi.org/10.1186/s1366201919462