Isotone maps on lattices

TL;DR

This paper investigates how isotone maps from families of lattices to a common lattice can be extended to free products within a lattice variety, revealing limitations and special cases for partial lattices and related structures.

Contribution

It extends known results on isotone map extensions to free products in lattice varieties, including partial lattices and alternative order structures.

Findings

Isotone maps extend to free products in lattice varieties.

The extension property fails for free lattices on arbitrary partial lattices unless the codomain is complete.

Special cases for partial lattices and related structures are identified.

Abstract

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.