The κ-word problem over DRH

Abstract

Profinite techniques are explored in order to prove decidability of a word problem over a family of pseudovarieties of semigroups, which is parameterized by pseudovarieties of groups. Let κ be the signature that naturally generalizes the usual signature on groups: it consists of the multiplication, and of the (ω − 1)-power. Given a pseudovariety of groups H, we denote by DRH the pseudovariety all finite semigroups whose regular R-classes lie in H. We prove that the word problem for κ-terms is decidable over DRH provided it is decidable over H (in general, the word problem for κ-terms is said to be decidable over a pseudovariety V if it is decidable whether two κ-terms define the same element in every semigroup of V). Further, we present a canonical form for elements in the free κ-semigroup over DRH, based on the knowledge of a canonical form for elements in the free κ-semigroup over H. This extends work of Almeida and Zeitoun on the pseudovariety of all finite R-trivial semigroups.