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Abstract: The classical Poincaré recurrence theorem in a probability space states that for any and any measure preserving map almost every point of A returns to A infinitely many times [9]. Here variations of the theorem are presented inspired by MV-algebras. [Copyright &y& Elsevier]

There is given a construction of the entropy of a dynamical system on arbitrary MV-algebra M. If M is the MV-algebra of characteristic functions of a σ-algebra (isomorphic to the σ-algebra), then the construction leads to the Kolmogorov–Sinaj entropy. If M is the MV-algebra (tribe) of fuzzy sets, then the construction coincides with the Maličký modification of the Kolmogorov–Sinaj entropy for fuzzy sets (Maličký and Riečan, 1986; Riečan and Mundici, 2002; Riečan and Neubrunn, 1997). [ABSTRACT FROM AUTHOR]

As an alternative to quantum logics MV algebras have been suggested[sup 2,20]. In a quite general class of MV algebras there have been constructed sum of any two observables[sup 10,14,20]. Recently the notion of product MV algebra has been introduced[sup 4,5,12,17]. In any weakly σ-distributive product MV algebra the joint observable of any observables can be constructed and on the base also the sum of any observables. In this paper we show that the two concepts of constructing of the sum are equivalent. [ABSTRACT FROM AUTHOR]

The present paper is aimed at studying the entropy of dynamical systems in product MV-algebras. First, by using the concept of logical entropy of a partition in a product MV-algebra 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 MV-algebra 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 MV-algebra dynamical systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput.2000, 4, 41–44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback–Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback–Leibler divergence in a product MV-algebra are examined. [ABSTRACT FROM AUTHOR]