In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all J-trivial (resp. R-trivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κ-tame when V is a subpseudovariety of J with decidable κ-word problem and W is...

In this paper we prove that, if V is a κ-tame pseudovariety which satisfies the pseudoidentity xyz = 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, LS...

The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg [1] as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties. Recall that the join V ∨ W of two pseudovarieties V and W is the least pseudovariety containing both V and W. This talk is concerned with the problem of proving tameness of joins. This problem was consider...

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 de...

This work gives a new approach to the construction of implicit operations. By considering “higher-dimensional” spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, ...

In this paper; we study equations on semidirect products of commutative semigroups. Let Comq,r denote the pseudovariety of all finite semigroups that satisfy the equations xy = yx and x r + q = xr. The pseudovariety Com1,1 is the pseudovariety of all finite semilattices. We consider the product pseudovariety Comq,r * generated by all semidirect products of the form S*T with S ∈ Comq,r and T ∈ ,...

This article answers three questions of J. Almeida. Using combinatorial, algebraic and topological methods, we compute joins involving the pseudovariety of finite groups, the pseudovariety of semigroups in which each idempotent is a right zero and the pseudovariety generated by monoids M such that each idempotent of M\{1} is a left zero.