Conjectures for the existence of an idempotent in $\omega $-polynomial algebras

Abstract

The existence of idempotent elements in baric algebras defined by $\omega$-polynomial identities ($\omega$-PI algebras) is an important problem for the study of genetic algebras. We conjecture here two criteria on the existence of an idempotent. These criteria are based on the existence of 1/2 as double root of a polynomial built from the identity defining a $\omega$-PI algebra. We show that these criteria are true in all the algebras studied until now and for which we have results concerning the existence of idempotent elements.