On the variety of strict pseudosemilattices

A new model, in terms of finite bipartite graphs, of the free pseudosemilattice is presented. This will then be used to obtain several results about the variety SPS of all strict pseudosemilattices: (i) an identity basis for SPS is found, (ii) SPS is shown to be inherently non-finitely based, (iii) SPS is shown to have no irredundant identity basis, and (iv) SPS is shown to have no covers and to be ∩-prime in the lattice of all varieties of pseudosemilattices. Some applications to e-varieties of locally inverse semigroups are also derived.

ALMOST FACTORIZABLE LOCALLY INVERSE SEMIGROUPS

We introduce a notion of almost factorizability within the class of all locally inverse semigroups by requiring a property of order ideals, and we prove that the almost factorizable locally inverse semigroups are just the homomorphic images of Pastijn products of normal bands by completely simple semigroups.

Ideal extensions of locally inverse semigroups

The translational hull of a locally inverse semigroup has a largest locally inverse subsemigroup containing the inner part. A construction is given for ideal extensions within the class of all locally inverse semigroups.