Forcing extensions of partial lattices

Details Forcing extensions of partial lattices Forcing extensions of partial lattices

2005-01-22

arxiv.org/abs/math/0501378v1

Journal of Algebra 262, no. 1 (2003) 127--193

Mathematics

General Mathematics

Scientific paper

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let $\phi$: 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 = $\phi$. 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 $\phi$ 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.

Affiliated with

Also associated with

No associations

Rating

Forcing extensions of partial lattices does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Forcing extensions of partial lattices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Forcing extensions of partial lattices will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-279858