The Ramsey and the ordering property for classes of lattices and semilattices

TL;DR

This paper investigates the Ramsey property in classes of finite lattices and semilattices, revealing that most such classes do not have the property unless viewed as ordered posets, where they do.

Contribution

It demonstrates that the lack of the Ramsey property in lattice classes is due to algebraic morphisms, and shows that ordered posets derived from these classes do possess the property.

Findings

Finite distributive lattices do not have the Ramsey property.

Expanding lattices with linear orders generally does not yield a Ramsey class.

Ordered posets derived from lattice classes do have the Ramsey and ordering properties.

Abstract

The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki\'c have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the variety of distributive lattices is not an exception, but an instance of a more general phenomenon. We show that for almost all nontrivial locally finite varieties of lattices no "reasonable" expansion of the finite members of the variety by linear orders gives rise to a Ramsey class. The responsibility for this lies not with the lattices as structures, but with the lack of algebraic morphisms: if we consider lattices as partially ordered sets (and thus switch from algebraic embeddings to embeddings of relational structures) we show that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property. It now comes as no surprise that the same is true for varieties of semilattices.