2005/01/22 by Friedrich Wehrung
Mathematics · #math.GM #msc:06B10 #msc:06B15 #msc:06B25 #msc:03C90
published as Journal of Algebra 262, no. 1 (2003) 127--193
arxiv created 2005/01/22 · arxiv updated 2009/12/01
We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ: Con K → 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 → L, and an isomorphism \ga: Conc L → 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.