Abstract
Let P be a configuration, i.e., a finite poset with top element. Let Forb(P) be the class of bounded distributive lat- tices L whose Priestley space P(L) contains no copy of P.W e show that the following are equivalent: Forb(P) is first-order de- finable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes member- ship in 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 ForbH(P) being a variety, or even a quasivariety.
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