On dibaric and evolution algebras

TL;DR

This paper characterizes dibaric algebras through ideals and introduces bq-homomorphisms, establishing criteria for dibaric and conservative algebras, and applies these concepts to evolution algebras of bisexual populations.

Contribution

It introduces bq-homomorphisms and criteria for dibaric and conservative algebras, and applies these to evolution algebras of bisexual populations.

Findings

Characterization of dibaric algebras via ideals

Introduction of bq-homomorphisms and their role

Conditions for evolution algebra of bisexual populations

Abstract

We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps $f, g$ of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair $(f,g)$ we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a linear operator. Moreover, some properties of dibaric algebra homomorphisms of such algebras are studied.