TAMENESS OF JOINS INVOLVING THE PSEUDOVARIETY OF LOCAL SEMILATTICES J. C. COSTA Centro de Matem´ atica e Departamento de Matem´ atica e Aplica¸c˜ oes Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal Email: jcosta@math.uminho.pt C. NOGUEIRA Escola Superior de Tecnologia e Gest˜ ao, Instituto Polit´ ecnico de Leiria Campus 2, Morro do Lena, Alto Vieiro, 2411-901 Leiria, Portugal Email: conceicao.veloso@ipleiria.pt July 2, 2012 In this paper we prove that, if V is a κ-tame pseudovariety which satisfies the pseudoiden- tity xy ω+1 z = xyz, then the pseudovariety join LSl ∨ V is also κ-tame. Here, LSl denotes the pseudovariety of local semilattices and κ denotes the implicit signature consisting of the multiplication and the (ω − 1)-power. As a consequence, we deduce that LSl ∨ V is decidable. In particular the joins LSl ∨ Ab, LSl ∨ G, LSl ∨ OCR and LSl ∨ CR are decidable. Keywords: Semigroup, local semillatice, tame pseudovariety, join of pseudovarieties, pseudo- word, graph equation system Mathematics Subject Classification 2000: 20M05, 20M07, 20M35 1 Introduction A pseudovariety of semigroups is a class of finite semigroups closed under taking subsemi- groups, homomorphic images and finite direct product. It is said to be decidable if its mem- bership problem has a solution, that is, if there is an algorithm to test whether a given finite semigroup lies in that pseudovariety. The join V ∨ W of two pseudovarieties V and W is the least pseudovariety containing both V and W. A well-known result of Albert, Baldinger and Rhodes [1] states that the join of two decidable pseudovarieties may not be decidable. Decidability also fails to be preserved by some other common pseudovariety operators, such as semidirect product, block product, Mal’cev product and power [22, 12]. An idea which has been recently explored by several authors is to impose stronger prop- erties on the pseudovarieties upon which the operators are to be applied that guarantee that the resulting pseudovarieties are decidable [3, 23]. When it was introduced, by Almeida and Steinberg [4], the notion of tameness seamed to be the suitable property for the case of the semidirect product operator, but that assertion is still not proved. However, many pseudova- rieties obtained from tame pseudovarieties using for instance the join operator are expected