Geodesic
Combinatorial trees in Priestley spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 2, pp. 217-234.

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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.
Classification : 03C05, 06A11, 06D20, 06D50, 06D55, 54F05
Keywords: distributive lattice; Priestley duality; poset; first-order definable 
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     title = {Combinatorial trees in {Priestley} spaces},
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Ball, Richard N.; Pultr, Aleš; Sichler, Jiří. Combinatorial trees in Priestley spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 2, pp. 217-234. http://geodesic.mathdoc.fr/item/CMUC_2005__46_2_a1/