Characterization of posets of intervals. (English) Zbl 1047.06002

Lihová, Judita

Arch. Math., Brno 36, No. 3, 171-181 (2000).

Let \(\operatorname {Int} A\) be the system of all nonempty intervals of a partially ordered set \(A\), ordered by inclusion. For each class \({\mathcal A}\) of partially ordered sets we denote by \({\mathcal P}({\mathcal A})\) the class of all partially ordered sets \(P\) having the property that there is a partially ordered set \(A_P\) such that \(\operatorname {Int} A_P\) is isomorphic to \(P\). In the present paper, the author gives an algebraic characterization of partially ordered sets in \({\mathcal P}({\mathcal A})\) for \({\mathcal A}\) being (1) the class of all bounded posets and (2) the class of all posets \(A\) satisfying the condition that for each \(a\in A\) there exist a minimal element \(u\) and a maximal element \(v\) with \(u\leq a\leq v\), respectively.

Reviewer: Jan Paseka (Brno)

MSC:

06A06

Partial orders, general

Keywords:

partially ordered set; interval of a partially ordered set; interval poset