Reducibility versus definability for pseudovarieties of semigroups

Abstract

It is easy to show that a pseudovariety which is reducible with respect to an implicit signature $\sigma$ for the equation $x=y$ can also be defined by $\sigma$-identities. We present several negative examples for the converse using signatures in which the pseudovarieties are usually defined. An ordered example issue from the extended Straubing-Th\'erien hierarchy of regular languages is also shown to provide a positive example for the inequality $x\le y$.