Existence varieties of locally inverse semigroups vs varieties of pseudosemilattices
by
Luís Oliveira
Marquette University, Milwaukee and University of Porto, Portugal

An existence variety (or, briefly, e-variety) of regular semigroups is a class of these algebras closed for homomorphic images, regular subsemigroups, and direct products. The class LI of all locally inverse semigroups is an example of an e-variety. Locally inverse semigroups can be characterized as regular semigroups where the sandwich set S(e, f)=fV(ef)e of any two idempotents e and f has exactly 1 element. A pseudosemilattice is an idempotent binary algebra (E(S), /\ ) where S is a locally inverse semigroup, E(S) is the set of idempotents of S, and f /\ e is defined as the unique element of S(e, f). Usually, pseudosemilattices are not semigroups.

To each e-subvariety V of LI, we can associate the class E_V of all pseudosemilattices (E(S), /\ ) such that S belongs to V. This association defines a complete surjective homomorphism from the lattice of e-subvarieties of LI to the lattice of subvarieties of PS, where PS designates the variety of all pseudosemilattices. We will address some questions about the structure of the lattice of varieties of pseudosemilattices (cardinality, covers, ...) which will lead to some results concerning on the lattice of e-varieties of locally inverse semigroups.

Date received: April 29, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajl-02.