Tameness of The Pseudovariety Ls1

A class of finite semigroups V is said to be decidable if the membership problem for V has a solution, that is, if we can construct an algorithm to test whether a given semigroup lies in V. Decidability of pseudovarieties is not preserved by some of the most common pseudovariety operators, such as semidirect product, Mal’cev product and join [1, 17]. In particular Rhodes [17] has exhibited a decidable pseudovariety V such that the semidirect product Sl ∗V is not decidable, where Sl is the pseudovariety of semilattices. This fact suggested the necessity to establish and to study stronger conditions which were expected to be important in proving decidability of pseudovarieties using these operators. In this context several concepts were introduced such as the notion of tameness, introduced by Almeida and Steinberg [7]. Tameness consists of a refinement of the concept of hyperdecidability previously introduced by Almeida [3]. If tameness is preserved by the most common operators is an open problem yet. That it is a property of the “usual” pseudovarieties is a conjecture. In general, proving tameness of a pseudovariety is a non trivial, but useful, exercise. Some pseudovarieties are already known to be tame such as, for example: G, the pseudovariety of all finite groups (consequence of the interpretation of Ash’s results [10]); Ab [6], the pseudovariety of all finite abelian groups; K and D [9], the pseudovarieties of semigroups whose idempotents are, respectively, left zeros and right zeros. For more examples and recent developments on the subject, the reader is referred to Almeida’s papers [4, 5]. Here we study the tameness of LSl, the pseudovariety of finite semigroups S such that eSe ∈ Sl, for all idempotents e ∈ S. This problem was proposed by Almeida