Crit\`ere d'existence d'idempotent bas\'e sur les alg\`ebres de R\'etrocroisement

TL;DR

This paper explores the existence of idempotent elements in backcrossing algebras, linking algebraic structures with genetic interpretations, and provides criteria for their existence under certain conditions.

Contribution

It establishes a criterion for the existence of idempotents in baric algebras with polynomial identities containing backcrossing subalgebras.

Findings

Every element of weight 1 in a backcrossing algebra generates a mutation algebra.

For any polynomial identity, a corresponding backcrossing algebra exists.

Criteria for idempotent existence in specific baric algebras are provided.

Abstract

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying $\omega$-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity $f$ there is a backcrossing algebra satisfying $f$. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.