Evolution algebra of a bisexual population

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.

Constrained evolution algebras and dynamical systems of a bisexual population

“…Following [9] we give an algebra structure on the vector space R n+ν which is closely related to the map defined by (2.2).…”

“…The algebra B = B V generated by the evolution operator V (see (2.2)) is called the evolution algebra of the bisexual population (EABP) [9]. In [9] the basic properties of the EABP are studied.…”

“…This algebra was introduced in [9] and the sets of idempotent and absolute nilpotent elements of the algebra B 1 are described. In this paper we continue the study of this algebra and corresponding dynamical systems.…”

“…Recently, in [9] a bisexual population was considered and an EA (denoted by B) using inheritance coefficients of the population was introduced. This algebra is a natural generalization of the algebra A of free population.…”

“…For such algebras sets of idempotent and absolute nilpotent elements are known [9]. In Section 2 we give necessary definitions.…”