Regraphs and congruence lattices
In an earlier paper [Algebra Univers. 10, 74-95 (1980; Zbl 0433.06009)] the authors introduced special graphs, called regraphs. Here they use them for studying the finite congruence lattice representation problem. An embedding \(\phi\) of a finite lattice L into the lattice Eq(A) of equivalences on a finite set A is algebraic if \(\phi\) preserves \(0_ L\) and \(1_ L\) and there exists an algebra \({\mathcal A}=(A,F)\) such that \(Im(\phi)=Con({\mathcal A})\). In this case L is representable as the congruence lattice of \({\mathcal A}\). The main result is the following one: If G is a perfect regraph valued by A and \(\phi\) an algebraic embedding of L into Eq(A), then the factorized G-power of \(\phi\) is algebraic (for corresponding notions see the cited paper). As a corollary one obtains: For every finite nontrivial lattice L there exists a positive integer \(n_ 0\) such that if L can be represented as a congruence lattice of a finite algebra, then it is possible to find such a representation on every set A with \(| A| \geq n_ 0\).
finite congruence lattice representation problem
scientific article; zbMATH DE number 3859187