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 $ϕ$: Con K $\to$ D be a {∨, 0}-homomorphism, where Conc K denotes the {∨, 0}-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f : K $\to$ L, and an isomorphism $\ga$: Conc L $\to$ D such that $\ga$ Conc f = $ϕ$. 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: -- Every lattice K such that Conc K is a lattice admits a congruence-preserving extension into a relatively complemented lattice. -- 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.