Representing a monotone map by principal lattice congruences

Abstract

For a lattice L, let Princ (L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Gratzer proved that bounded ordered sets (in other words, posets with 0 and 1) are, up to isomorphism, exactly the Princ (L) of bounded lattices L. Here we prove that for each 0-separating boundpreserving monotone map ψ between two bounded ordered sets, there are a lattice L and a sublattice K of L such that, in essence, ψ is the map from Princ (K) to Princ (L) that sends a principal congruence to the congruence it generates in the larger lattice.