Evolution algebra of a "chicken" population

TL;DR

This paper investigates the structural properties of an evolution algebra modeling a bisexual chicken population, including conditions for associativity, nilpotency, and classification of low-dimensional cases.

Contribution

It introduces a new algebraic framework for bisexual populations and characterizes its properties, including classifications of low-dimensional cases and conditions for various algebraic identities.

Findings

The algebra is commutative but not associative or unital.

Conditions are identified for associativity, alternative, and power associative properties.

Complete classifications of 2D and some 3D algebras are provided.

Abstract

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.