Graphs, pseudosemilattices and idempotent generated locally inverse semigroups

The product of two idempotents of a locally inverse semigroup is not idempotent in general. There is, however, another operation that can be defined naturally using the semigroup operation such that the product of two idempotents is an idempotent. These new algebras are called pseudosemilattices (they generalize the notion of semilattice) and they are not semigroups in general. We shall begin this talk by recalling some results about the relation between pseudosemilattices and idempotent generated locally inverse semigroups. We shall then introduce a special type of graphs that shall be used to construct models for the free pseudosemilattices and the free idempotent generated locally inverse semigroups. The representation of the elements of these models resembles the Munn tree representation of the elements of the free inverse semigroup.