Reducibility vs. definability for pseudovarieties of semigroups
J. Almeida, Ondřej Klíma · International Journal of Algebra and Computation · 2016
It is easy to show that a pseudovariety which is reducible with respect to an implicit signature [Formula: see text] for the equation [Formula: see text] can also be defined by [Formula: see text]-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–Thérien hierarchy of regular languages is also shown to provide a positive example for the inequality [Formula: see text].