The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

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 Sokić 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.

Dual Ramsey properties for classes of algebras

“…For example, it was shown in [19] that no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property (although there is an expansion using ternary relations, see [19]). Motivated by this result in [24] we show that for an arbitrary nontrivial locally finite variety V of lattices distinct from the variety of all lattices and the variety of distributive lattices, no reasonable expansion of V fin (= the class of all the finite lattices in V) has the Ramsey property. However, if we consider lattices as partially ordered sets (and thus switch from the lattices as algebras to their relational alter ego) we show in [24] that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property (see [29] for definition).…”

“…Motivated by this result in [24] we show that for an arbitrary nontrivial locally finite variety V of lattices distinct from the variety of all lattices and the variety of distributive lattices, no reasonable expansion of V fin (= the class of all the finite lattices in V) has the Ramsey property. However, if we consider lattices as partially ordered sets (and thus switch from the lattices as algebras to their relational alter ego) we show in [24] that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property (see [29] for definition). It seems that structural Ramsey theory is at odds with natural (equationally defined) classes of finite algebras.…”