Pseudosemilattices are idempotent algebras that generalize the notion of semilattice. Although these algebras are not semigroups in general (the binary operation is generally non-associative), they derive from a particular class of semigroups, the locally inverse semigroups. On each locally inverse semigroup (S,●) it is possible to introduce naturally a new binary operation ᴧ on the set E(S) of idempotents of S. The new binary algebra (E(S), ᴧ) is a pseudosemilattice and each pseudosemilattice can be obtained in this way. The class of all pseudosemilattices constitutes a variety of algebras. In this talk we shall present a recently found model for the free pseudosemilattice on a set X. The elements of this model are special graphs on which we shall define a binary operation ᴧ . We shall present some results about pseudosemilattices and locally inverse semigroups obtained from the analysis of this model. The results presented in this talk were partially obtained in a joint work with Karl Auinger.