Regraphs and congruence lattices

Abstract

We shall investigate embeddings ~p :L-* Eq(A) of a finite lattice L into the lattice of all equivalences over a finite set A*. Such an embedding is called complete, if it preserves the least and the greatest e lements 0L and l c of L. Let q~ : L ~ Eq(A) be a complete embedding. Then we say that q~ is algebraic if there is an algebra ~ = (A, F) such that its congruence lattice Con(~) is equal to Ira(q0 (the image of ~0). In this case we say also that L is representable as the congruence lattice of the algebra sr It is an open problem whether each finite lattice is representable as the congruence lattice of a finite algebra. It is easily seen that without loss of generality we can consider only algebras (A, F) where F is a monoid of unary operat ions on A. A regraph valued by a set A is a triple (G, R, o-) = G, where G is a non-empty set, R is a symmetr ic antireflexive relation on G, and ~r is a mapping of R into A. If, moreover , a mapping q~:L--~ Eq(A) is given, we define a new mapping : L ~ Eq(A x G), called G-power of q~ as follows: for x 6 L, qJ(x) is the least equivalence on A • G containing the following two relations