Tsallis Entropy of Product MV-Algebra Dynamical Systems

Abstract

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.

Tsallis Entropy of Fuzzy Dynamical Systems

“…It can be verified that the function L q is, for every q ∈ (0, 1) ∪ (1, ∞), concave and non-negative. The non-negativity of the function L q (for the proof, see [56]) implies that the Tsallis entropy is always non-negative. Obviously, by inserting q = 2 into Equation 11, we obtain the logical entropy H L (α).…”

“…Proof. The proof can be done using part (i) of Proposition 1 and Proposition 3, in the same way as the proof of Theorem 3 in Reference [56].…”

“…[53][54][55]) was defined and studied in Reference [10]. Another recently published paper [56] was devoted to the mathematical modeling of Tsallis entropy in product MV-algebra dynamical systems. It should be noted that the full tribe of fuzzy sets represents a special case of product MV-algebras; therefore, the results provided in References [10,56] can be immediately applied to this important case of fuzzy sets.…”

“…Another recently published paper [56] was devoted to the mathematical modeling of Tsallis entropy in product MV-algebra dynamical systems. It should be noted that the full tribe of fuzzy sets represents a special case of product MV-algebras; therefore, the results provided in References [10,56] can be immediately applied to this important case of fuzzy sets. It is known that there are many possibilities for defining operations over fuzzy sets; for an overview, see Reference [57].…”

Entropy in Dynamic Systems

“…Tsallis Entropy of Product MV-Algebra Dynamical Systems by Dagmar Markechová and Beloslav Riečan [ 5 ] provides an example of the mathematical modelling of Tsallis of product MV-algebra dynamical entropy to provide the entropy measure that is invariant under isomorphism.…”