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 ${\bf LG}$ of all finite local groups and use them as test-semigroups to prove that ${\bf LG}$ and ${\bf S}$, 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.

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