Weak congruence lattice representation problem
by
Branimir Seselja
University of Novi Sad
Coauthors: Andreja Tepavcevic (University of Novi Sad)

Formulation of the problem: If L is an algebraic lattice and p its element, find an algebra A whose weak congruence lattice is isomorphic with L, so that p corresponds to the diagonal relation on A.

The problem has not been generally solved, but we give positive answers in some cases and present several new results. In one hand, there is a list of conditions which should be satisfied by an element of L corresponding to the diagonal relation of an algebra which represents L by weak congruences. Further, some properties of L uniquely determine particular algebraic properties of each algebra representing L. Among these are Hamiltonian property, regularity of congruences, CEP, existence of nullary operations, CIP, subalgebras as blocks of congruences and others. Therefore, starting with a lattice L and its element p, the following answers to the above problem can be given:

-there is a representation (in some cases);

-there is no representation;

-if there is an algebra A representing L, then A must satisfy several algebraic properties (from the given list).

Complete answers to the above questions are given for all lattices whose order is less than 7. In each case an algebra representing the lattice is constructed. Except in two cases, these algebras are finite.

Date received: July 15, 1999