In order to solve the k-word problem for the pseudovariety LG of local groups, we began constructing examples of local groups to separate some k-terms. This naturally guided us to the construction of a family of generators of LG, the called test-semigroups for LG, and, in parallel, to the definition of a canonical form of k-terms over LG.

In this talk we introduce a type of semigroup presentations and show that the semigroups it defines is the referred family of test-semigroups. We will also show that the free objects of the semidirect products of the form G*Dn ($n\in\mathbb N$) are subsemigroups of members of an enlarged family of local groups defined by a type of semigroup presentations which generalizes the previous one