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In this paper, we connect four different branches of Mathematics: Statistics, Probability, Algebra and Discrete Mathematics with the objective of introducing new results on Markov chains and evolution algebras obtained by following a relatively new line of research, already dealt with by several authors. It consists of the use of certain directed graphs to facilitate the study of Markov chains and evolution algebras, as well as to use each of the three objects to make easier the study of the other two. The results obtained can be useful, in turn, to link different scientific disciplines, such as Physics, Engineering and Mathematics, in which evolution algebras are considered very interesting tools. [ABSTRACT FROM AUTHOR]

This paper deals with the evolution operator of evolution algebras. The main goal is to obtain novel results of this operator, both in the case in which it is a derivation and when it is a homomorphism of algebras. These new results constitute a criterion for classifying evolution algebras satisfying one of these conditions. An algorithm to obtain all these degenerate evolution algebras starting from those of smaller dimensions is also constructed. [ABSTRACT FROM AUTHOR]

Although since their introduction by Tian in 2004, evolution algebras have been the subject of a very deep study in the last years due to their numerous applications to other disciplines, this study is not easy since these algebras lack an identity that characterizes them, such as the identity of Jacobi, for Lie algebras or those of Leibniz and Malcev for those corresponding algebras. In this paper we deal with the concepts of solvability and nilpotency of these evolution algebras. Some novel results on them obtained from using the evolution operator of these algebras are given and some examples illustrating these results are also shown. The main result obtained states that an evolution algebra is solvable if and only if its structure matrix is nilpotent, which implies, in turn, that the solvability and the nilpotency indices of that algebra coincide provided the corresponding evolution operator is an endomorphism of the algebra. [ABSTRACT FROM AUTHOR]

This article deals with the evolution operator of evolution algebras. We give a theorem that allows to characterize these algebras when this operator is a homomorphism of algebras of rank n−2 and this result in turn allows us to extend the classification of this type of algebras, given in a previous result by ourselves in 2021, up to the case of dimension 4. For this purpose, we analyze and make use of an algorithm for the degenerate case. A computational study of the procedure is also made. [ABSTRACT FROM AUTHOR]

We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras. [ABSTRACT FROM AUTHOR]

In this paper, we firstly study a commutative algebra E over a field F of Char(F) ⵐ 2 that satisfying dim(Ꜫ ² ) = 1. We show that, such an algebra is an evolution algebra. Afterwards, we pay attention to commutative duplicate of a commutative algebra E. We find necessary and sufficient condition in which the duplicate D(E) is an evolution algebra. And, we finish by studying an evolution algebra that is a Bernstein algebra. We classify that algebras, up to isomorphism, in dimension ≤ 4. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new results on evolution algebras obtained by following a relatively new line of research on these objects. It consists of the use of certain properties of graphs to facilitate the study of evolution algebras and reciprocally. [ABSTRACT FROM AUTHOR]

Evolution algebras are currently widely studied due to their importance not only “per se” but also for their many applications to different scientific disciplines, such as Physics or Engineering, for instance. This paper deals with these types of algebras and their applications. A criterion for classifying those satisfying certain conditions is given and an algorithm to obtain degenerate evolution algebras starting from those of smaller dimensions is also analyzed and constructed.

Through this paper, we show that the criteria for real evolution algebra to be a baric algebra can be extended to any evolution algebra over a commutative field of characteristic 6 ≠ = 2. Then we prove that an evolution algebra E is a train algebra of rank r + 1 if and only if the kernel of its weight function is nil of nil-index r > 1. We also study special train evolution algebra and characterize idempotents, power-associativity and automorphism in evolution train algebra. Finally we classify up to dimension 5, indecomposable evolution nil-algebra of nil-index 4 that are not power-associative. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, initiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. [ABSTRACT FROM AUTHOR]

To use evolution algebras to model population dynamics that both allow extinction and introduction of certain gametes in finite generations, nilpotency must be built into the algebraic structures of these algebras with the entire algebras not to be nilpotent if the populations are assumed to evolve for a long period of time. To adequately address this need, evolution algebras over rings with nilpotent elements must be considered instead of evolution algebras over fields. This paper develops some criteria, which are computational in nature, about the nilpotency of these algebras, and shows how to construct finitely generated evolution algebras which are nil but not nilpotent. [ABSTRACT FROM AUTHOR]

Abstract: This study constitutes the continuation of innovative research in Discrete Mathematics introduced in earlier papers on algebras in general, regarding the use of graphs to study the particular case of graphicable algebras, which form a subset of evolution algebras. Evolution algebras are particularly interesting since they are intrinsically linked with other mathematical fields, such as group theory, stochastics processes, and dynamical systems, for instance. Our advances in this study are obtained by setting a natural correspondence between evolution algebras and direct graphs, in order to translate the general concepts of graphicable algebras: subalgebra, ideal, centralizer, normalizer…to the language of graphs. These translations will enable advances in the application of these algebras to various branches of Mathematics. [Copyright &y& Elsevier]

In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure µ. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces has a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras. [ABSTRACT FROM AUTHOR]

We extend the concept of the "join" to the case of infinitely many weighted algebras. We study the problem of its uniqueness (up to weighted isomorphism) which gives rise to a natural notion of homogeneous weighted algebras. We show that several classes of weighted algebras coming from genetics are homogeneous and that homogeneity is preserved by duplication. Finally, we examine some well-known weighted algebras satisfying identities, as Bernstein, train, and evolution algebras. [ABSTRACT FROM AUTHOR]