**José Carlos Costa**

*CMAT, University of Minho*

An inverse semigroup is a regular semigroup whose idempotents commute. A remarkable result obtained by Ash in 1987 is that the finite inverse semigroups generate the pseudovariety ECom of idempotent commuting semigroups. Somehow surprisingly, Higgins and Margolis have shown in 2000 that the pseudovariety AInv generated by finite aperiodic inverse semigroups is properly contained in the pseudovariety AECom of aperiodic semigroups with commuting idempotents.

A kappa-term is a formal expression obtained from the letters of an alphabet using the operations of multiplication and (omega-1)-power. I will talk about the word problems for kappa-terms over the pseudovarieties AInv, AECom and ECom. The decidability of these problems is proven through automata associated with kappa-terms.

This is a joint work with Mário Branco (Universidade de Lisboa), Conceição Nogueira (Instituto Politécnico de Leiria) and M. Lurdes Teixeira (Universidade do Minho).