Isotone maps on lattices

Details Isotone maps on lattices Isotone maps on lattices

2011-03-07

arxiv.org/abs/1103.1339v1

Mathematics

Rings and Algebras

Scientific paper

Let (L_i : i\in I) be a family of lattices in a nontrivial lattice variety V, and let \phi_i: L_i --> M, for i\in I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps \phi_i can be extended to an isotone map \phi: L --> M, where L is the free product of the L_i in V. This was known for V the variety of all lattices (Yu. I. Sorkin 1952). The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the L_i. The analog of the above result does not, however, hold for the free lattice L on an arbitrary partial lattice P. We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lattices:orders replaced either by semilattices:orders or by lattices:semilattices. Some open questions are noted.

Affiliated with

Also associated with

No associations

Rating

Isotone maps on 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 Isotone maps on lattices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Isotone maps on lattices will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-607644