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

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
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