Semigroup presentations for test local groups

In this paper we exhibit a type of semigroup presentation 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.