The semidirectly closed pseudovariety generated by aperiodic Brandt semigroups

Abstract

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup $B_2$, denoted $\mathbf V^*(B_2)$. We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by $B_2$ which leads to the main result, which states that $\mathbf V^*(B_2)$ is decidable. Independently, using some suggestions given by J.~Almeida in his book ``Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in $\mathbf V^*(B_2)$.