The concept of tameness of a
pseudovariety was introduced recently by Almeida and
Steinberg [1], who established the following result.

Let be the pseudovariety of all finite semilattices. This talk is concerned with the tameness of the pseudovariety of all finite semigroups such that for each idempotent . Recall that the pseudovariety is associated through Eilenberg's correspondence with the variety of locally testable languages, and that is the semidirect product where stands for the pseudovariety of all finite semigroups in which idempotents are right zeros. The proof of the tameness of is a problem proposed by Almeida [2].

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*.