Markov evolution algebras

Abstract

We study Markov evolution algebras, that is, evolution algebras having Markov structure matrices. We first consider the discrete-time case, and delve into their algebraic structure for later application to continuous-time Markov evolution algebras that arise defined by standard stochastic semigroups. The study of embeddable Markov evolution algebras, that is, of Markov evolution algebras with structure matrices existing within standard stochastic semigroups is then equivalent to the embedding problem for Markov matrices.

The connection between evolution algebras, random walks and graphs

“…Some of the main lines of research are focused on the study of different mathematical properties like the classification of evolution algebras [1], the characterization of their derivation spaces [2,3,6,8], and the analysis of related co-algebras [21][22][23]. Regarding connections with other fields fields, one can refer the reader to the following works [4,5,10,20,24,26]. The link with Probability Theory is through the concept of discrete-time Markov chain, this was suggested first by [28,Chapter 4], were many properties of Markov chains were translated in the language of evolution algebras.…”

“…The link with Probability Theory is through the concept of discrete-time Markov chain, this was suggested first by [28,Chapter 4], were many properties of Markov chains were translated in the language of evolution algebras. Such connection has been explored in different ways, for example, by [9,20,24], and also inspired new lines of research like in [30]. The interplay between evolution algebras and dynamical systems is through the interpretation of the evolution operator of the algebra, which is defined as the endomorphism L : A → A, such that L(e i ) = e 2 i , for any i ∈ Λ.…”

On S-Evolution Algebras and Their Enveloping Algebras

“…Therefore, it is natural to investigate such algebra, which may open some shed into these games from an algebraic point of view. Note that evolution algebra associated with the Markov process has been considered in [8,15,45]. However, the considered evolution algebra is not Markov evolution algebra, and hence, it is needed for the investigation (algebraic properties ) Example 3.…”

Entropy Treatment of Evolution Algebras

“…In [25], Markov evolution algebras, whose stricture matrices obey semi-group property, were investigated. This type of study is related to the chain of evolution algebras [26].…”