A chain of evolution algebras

TL;DR

This paper introduces the concept of chains of evolution algebras, explores their properties, examples, and criteria for baric property, and analyzes their behavior over time with respect to idempotent and nilpotent elements.

Contribution

It defines chains of evolution algebras satisfying Chapman-Kolmogorov-like equations, provides examples, and studies properties like baricness and idempotents over time.

Findings

Periodic chains form a continuum of non-isomorphic algebras.

The baric property can transition over time depending on the chain.

The number of idempotent elements can change at a critical time.

Abstract

We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman-Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time $t_{c}$ such that the chain has only two idempotent elements if time $t\geq t_{\rm c}$ and it has four idempotent elements if time $t< t_{\rm c}$