On real chains of evolution algebras

Abstract

Abstract. In this paper we define a chain of n-dimensional evolution algebras corresponding to a permutation of n numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is trivial (consisting only algebras with zeromultiplication) iff the permutation has not a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) iff it is trivial. We construct a wide class of chains of 3-dimensional EAs and a class of symmetric n-dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of 3-dimensional EAs we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

Classifying Evolution Algebras of Dimensions Two and Three

“…In works such as [3][4][5][6][7][8][9][10] they studied purely mathematical notions, such as nilpotency and solvency of evolution algebras, as well as the interpretation of these mathematical notions, relating, for example, the nilpotency of an element to gametes that go extinct after some generations. Chains of evolution algebras were studied in [11][12][13][14]. These are dynamic systems where the state of each system can be seen as an evolution algebra.…”

An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

“…Each basis vector represents a genotype of a given phenotype so that the role of self-replication of genotypes is played by the products e i e i = ∑ n j=1 c ij e j , for all 1 ≤ i ≤ n. In the case of dealing with K = R as the base field, if ∑ n j=1 c ij = 1 and c i,j ≥ 0, for all i, j, then each structure constant c ij represents the probability that the genotype e i becomes e j in the offspring. In any case, even if there exist some advances in the algebraic study of such a probabilistic meaning [4,5], evolution algebras are usually studied without probabilistic restrictions on their structure constants [6][7][8][9][10][11][12][13][14][15][16].…”

Algebraic computation of genetic patterns related to three-dimensional evolution algebras

“…Evolution algebras were introduced by Tian and Vojtechovsky [22,23] to simulate algebraically the self-reproduction of alleles in non-Mendelian Genetics. In the last years, these algebras have been widely studied without probabilistic restrictions on their structure constants [3,6,10,8,13,17,19,18,21]. A main problem to be solved in this regard is the distribution of evolution algebras into isomorphism classes.…”