A constructive approach to the finite congruence lattice representation problem

Abstract

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. In this paper, we develop methods by which we can construct new representable lattices from known ones. The techniques we employ are sufficient to show that every finite lattice which contains no three element antichains is representable. We then show that if an order polynomially complete lattice is representable then so is every one of its diagonal subdirect powers.

Graphical algebras — a new approach to congruence lattices

“…In [2], M. Haiman's notion of the polynomial corresponding to a series-parallel graph is exactly the largest possible graphical composition, and is equal to the graphical composition in the case of commuting equivalence relations, which he was studying. In [9], [10] and [11], J. W. Snow used the same notion of graphical composition, described using certain types of logical statements, to produce ways of constructing new finite lattices with representations as the congruence lattice of a finite algebra, from old ones, and also to study hereditary congruence lattices, namely ones such that every sublattice is a congruence lattice on an algebra with more operations and the same underlying set.…”

Subdirect products of hereditary congruence lattices

“…In [5] the author proves that the set of all representable finite lattices is closed under certain lattice theoretic operations. We will employ some of those results here.…”

Every finite lattice in $$ \mathcal{V} (M_3) $$

“…In [5], the author exploits the following lemma which follows from the fact that a set of relations on a finite set is the set of all relations compatible with an algebra on the set if and only if the relations are closed under primitive positive definitions [1,2]. …”

“…In [5], the author investigates constructions by which one can create new representable lattices from lattices already known to be representable. In this paper, we use similar tactics to prove that any finite lattice in the variety generated by M 3 is representable.…”