On the Dual Ramsey Property for Finite Distributive Lattices

TL;DR

This paper investigates the dual Ramsey property for finite distributive lattices, showing it does not hold generally but establishing a dual Ramsey theorem for lattices with a specific linear order.

Contribution

It proves that the class of finite distributive lattices lacks the dual Ramsey property, but identifies conditions under which a dual Ramsey theorem applies.

Findings

Finite distributive lattices do not have the dual Ramsey property.

A dual Ramsey theorem is established for lattices with a particular linear order.

The general class of finite distributive lattices cannot be expanded to satisfy the dual Ramsey property.

Abstract

The class of finite distributive lattices, as many other classes of structures in everyday use, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriatelly 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 Sokic 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 class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order.