A new approach to the entropy of a transitive BE-algebra with countable partitions

Abstract

The concept of entropy and information gain of BE-algebras in scientific disciplines such as information theory, data science, supply chain and machine learning assists us to calculate the uncertanity of the scientific processes of phenomena. In this respect the notion of filter entropy for a transitive BE-algebra is introduced and its properties are investigated. The notion of a dynamical system on a transitive BE-algebra is introduced. The concept of the entropy for a transitive BE-algebra dynamical system is developed and, its characteristics are considered. The notion of equivalent transitive BE-algebra dynamical systems is defined, and it is proved the fact that two equivalent BE-algebra dynamical systems have the same entropy. Theorems to help calculate the entropy are given. Specifically, a new version of Kolmogorov– Sinai Theorem has been proved. The study introduces the concept of information gain of a transitive BE-algebra with respect to its filters and investigates its properties. This study proposes the use of filter entropy to approximate the level of risk introduced by a BE-algebra dynamical system. This aim is reached by defining the information gain with respect to the filters of a BE-algebra. This methodology is well developed for use in engineering, especially in industrial networks. This paper proposes a novel approach to assess the quantity of uncertainty, and the impact of information gain of a BE-algebra dynamical system.