Forcing extensions of partial lattices

✍ Scribed by Friedrich Wehrung

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 480 KB
• Volume: 262
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00015-2

✦ Synopsis

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ : Con c K → D be a {∨, 0}-homomorphism, where Con c 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 α :

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 Con c K 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.