Configurations in coproducts of Priestley spaces

Abstract

Let P be a configuration, i.e., a finite poset with top element. Let F(P) be the class of bounded distributive lattices L whose Priestley space P(L) contains no copy of $P$. The following are equivalent: F(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 F(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 onditions are also equivalent to $\fb(P)$ being a variety, or even a quasivariety.