Semigroup presentations for test local groups

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 LG of all finite local groups and use them as test-semigroups to prove that LG and S, the pseudovariety of all finite semigroups, verify the same κ-identities involving κ-terms of rank at most 1, where κ denotes the implicit signature consisting of the multiplication and the (ω − 1)-power.