Tameness of ${\cal L}{\bf Sl}$








J. C. Costa
Universidade do Minho, Portugal
L. Teixeira
Universidade do Minho, Portugal






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

Theorem 1   If ${\bf V}_1,\ldots,{\bf V}_n$ are tame pseudovarieties, then the semidirect product ${\bf V}_1*\cdots*{\bf V}_n$ is decidable.
Since then, the notion of tameness has received the attention of many authors being a number of works dedicated to proving tameness of pseudovarieties.

Let ${\bf Sl}$ be the pseudovariety of all finite semilattices. This talk is concerned with the tameness of the pseudovariety ${\cal L}{\bf Sl}$ of all finite semigroups $S$ such that $eSe\in {\bf Sl}$ for each idempotent $e\in S$. Recall that the pseudovariety ${\cal L}{\bf Sl}$ is associated through Eilenberg's correspondence with the variety of locally testable languages, and that ${\cal L}{\bf Sl}$ is the semidirect product ${\bf Sl}*{\bf D}$ where ${\bf D}$ stands for the pseudovariety of all finite semigroups in which idempotents are right zeros. The proof of the tameness of ${\cal L}{\bf Sl}$ 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.

 

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