An introduction of logical entropy on sequential effect algebra

2018

Mathematics

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

Section: Introductionmentioning

confidence: 75%

Tsallis Entropy of Fuzzy Dynamical Systems

2018

Mathematics

“…It has been proven that the logical entropy H l (T) distinguishes non-isomorphic 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 above-mentioned studies are possible in the Kolmogorov probability theory based on the modern integration theory.…”

Section: Introductionmentioning

confidence: 99%

Logical entropy of dynamical systems in product MV-algebras and general scheme

Riečan

2019

Adv Differ Equ

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.

“…The logical entropy

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 ].…”

Section: Introductionmentioning

confidence: 99%

Tsallis Entropy of Product MV-Algebra Dynamical Systems

Riečan

2018

Entropy

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.