Combinatorial trees in Priestley spaces. (English) Zbl 1121.06003

Ball, Richard N.; Pultr, Aleš; Sichler, Jiří

Commentat. Math. Univ. Carol. 46, No. 2, 217-234 (2005).

Summary: 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.

MSC:

06A11	Algebraic aspects of posets
54F05	Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
06D20	Heyting algebras (lattice-theoretic aspects)

Keywords:

distributive lattice; Priestley duality; poset