Isotone maps on lattices

Abstract

Let \({\mathcal{L} = (Li | i \in I)}\) be a family of lattices in a nontrivial lattice variety V, and let \({\varphi_{i} : L_{i} \rightarrow M}\), 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 \({\varphi_{i}}\) can be extended to an isotone map \({\varphi : L \rightarrow M}\), where \({L = {\rm Free}_{V} \mathcal{L}}\) is the free product of the Li in V. This was known for V = L, the variety of all lattices. 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 Li. 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.