Reducibility vs. 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-Therien hierarchy of regular languages is also shown to provide a positive example for the inequality x <= y.