Configurations in coproducts of Priestley spaces. (English) Zbl 1086.06012

Ball, Richard N.; Pultr, Aleš; Sichler, Jiří

Appl. Categ. Struct. 13, No. 2, 121-130 (2005).

Summary: Let \(P\) be a configuration, i.e., a finite poset with top element. Let \(\text{Forb}(P)\) be the class of bounded distributive lattices \(L\) whose Priestley space \({\mathcal P}(L)\) contains no copy of \(P\). We show that the following are equivalent: \(\text{Forb}(P)\) is first-order definable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes membership in \(\text{Forb}(P)\); \(P\) is coproductive, i.e., \(P\) embeds in a coproduct of Priestley spaces iff it embeds in one of the summands; \(P\) is a tree. In the restricted context of Heyting algebras, these conditions are also equivalent to \(\text{Forb}_{H}(P)\) being a variety, or even a quasivariety.

MSC:

06D50	Lattices and duality
06D20	Heyting algebras (lattice-theoretic aspects)
06D22	Frames, locales

Keywords:

distributive lattice; Priestley space; ultrafilter; coproduct