On a characterization of lattices by the betweenness relation — on a problem of M. Kolibiar. (English) Zbl 0757.06003

Hedlíková, Jarmila; Katriňák, Tibor

Algebra Univers. 28, No. 3, 389-400 (1991).

A ternary relation \(R\) on a lattice \(L\) is called a betweenness relation on \(L\) if \((a,b,c)\in R\) is defined by \((a\land b)\lor(b\land c)=b=(a\lor b)\land(b\lor c)\). M. Kolibiar [Z. Math. Logik Grundlagen Math. 4, 89–100 (1958; Zbl 0087.26002)] proved that a ternary relation \(R\) on a set \(L\) is the betweenness relation of some lattice structure on \(L\) iff it satisfies four conditions labeled (A), (B), (C), and (F). These conditions are explicitly described in the current paper, but are too complicated to reproduce here. The point is that Kolibiar proved the independence of the first three conditions, but left open the independence of (F) from the remaining conditions. In the current paper, the question is answered in the affirmative. There are also some interesting results on the question of whether the ternary betweenness relation on (distributive) lattices has a first-order axiomatization.

Reviewer: M. F. Janowitz (Amherst)

MSC:

06B05	Structure theory of lattices
03C07	Basic properties of first-order languages and structures

Keywords:

segment; convex sublattice; ternary relation; betweenness relation; first-order axiomatization

Citations:

Zbl 0087.26002

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