Semigroup presentations for test local groups

Abstract

In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety $\mathbf{LG}$ of all finite local groups and use them as test-semigroups to prove that $\mathbf{LG}$ and $\mathbf{S}$, the pseudovariety of all finite semigroups, verify the same $\kappa$-identities involving $\kappa$-terms of rank at most 1, where $\kappa$ denotes the implicit signature consisting of the multiplication and the $(\omega -1)$-power.