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distributive lattice; Priestley duality; poset; first-order definable
We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting $n$-crowns with $n\geq 3$ does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
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