Varieties of Semilattice-Ordered Semigroups
by
Martin Kuřil
J.E. Purkyně University, Ústí nad Labem, Czech Republic

We will present some results obtained jointly by Martin Kuril and Libor Pol\' ak.

A structure (A, ·, \/ ) is called a semilattice-ordered semigroup if

(i) (A, ·) is a semigroup

(ii) (A, \/ ) is a semilattice

(iii) for any a, b, c in A, a(b \/ c)=ab \/ ac and (a \/ b)c=ac \/ bc.

Libor Pol\' ak has generated interest in semilattice-ordered semigroups by establishing a connection with languages.

This contribution will be devoted to varieties of semilattice-ordered semigroups.

Let S be a free semigroup on the set {x₁, x₂, x₃, ... }, let \rho be a fully invariant congruence on S. A \rho-admissible closure operator is a closure operator on S/\rho satisfying some additional special conditions (we will specify them in our talk). Our main theorem asserts that there exists a one-to-one correspondence between all varieties of semilattice-ordered semigroups and all ordered pairs (\rho, [ ]) where \rho is a fully invariant congruence on S and [ ] is a \rho-admissible closure operator. Let V be a variety of semigroups. Denote by \rho_V the fully invariant congruence on S corresponding to the variety V. Let SLOV be the class of all semilattice-ordered semigroups (A, ·, \/ ) with the property (A, ·) in V. Our theorem reduces finding of all subvarieties in SLOV to the description of all \rho-admissible closure operators for all fully invariant congruences \rho on S such that \rho contains or equal \rho _V.

Recently, Stephen J. Emery has described the lattice of varieties of ordered normal bands. Using the theorem mentioned above we find all varieties of semilattice-ordered normal bands.

Date received: May 31, 2000

Copyright © 2000 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 # caec-35.