Geodesic
Combinatorial trees in Priestley spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 2, pp. 217-234 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Ball, Richard N. and Pultr, Ale\v{s} and Sichler, Ji\v{r}{\'\i}},
     title = {Combinatorial trees in {Priestley} spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {217--234},
     year = {2005},
     volume = {46},
     number = {2},
     mrnumber = {2176889},
     zbl = {1121.06003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2005_46_2_a1/}
}
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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/