Forcing extensions of partial lattices

Abstract

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ:ConcK→D be a {∨,0}-homomorphism, where ConcK denotes the {∨,0}-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f:K→L, and an isomorphism α:ConcL→D such that α∘Concf=ϕ. Furthermore, L and f satisfy many additional properties, for example: (i) L is relatively complemented; (ii) L has definable principal congruences; (iii) If the range of ϕ is cofinal in D, then the convex sublattice of L generated by f[K] equals L. We mention the following corollaries, that extend many results obtained in the last decades in that area: 1. Every lattice K such that ConcK is a lattice admits a congruence-preserving extension into a relatively complemented lattice. 2. Every {∨,0}-direct limit of a countable sequence of distributive lattices with zero is isomorphic to the semilattice of compact congruences of a relatively complemented lattice with zero.