Statisch pairs in atomistic posets. (English) Zbl 1424.06004

Vaezi, Alireza; Kharat, Vilas

Math. Bohem. 142, No. 2, 125-136 (2017).

The paper is devoted to the so-called statisch pairs in atomistic posets. Let \(P\) be an atomistic poset. An element a of \(P\) is called finite if either \(a=0\) or \(a\) belongs to the set of minimal elements of the set of upper bounds of the set of atoms below \(a\). A pair of elements \(a\), \(b\) is called statisch if for an atom \(p\) in \(\mathrm{UL}(a,b)\) there are finite elements \(c\) below \(a\) and \(d\) below \(b\) such that \(p\) is in \(\mathrm{UL}(c,d)\). A pair \(a,b\) is called a biatomic pair if for every atom \(p\) in \(\mathrm{UL}(a,b)\) there are atoms \(q\) below \(a\) and \(r\) below \(b\) such that \(p\) is in \(\mathrm{UL}(q,r)\).
The main results are: In an atomistic poset, dual modular, biatomic and statisch pairs coincide. The set of all finite elements of a statisch poset \(P\) forms an ideal of \(P\). A certain relation between standard elements is introduced and the authors study this relation by means of statisch pairs.

Reviewer: Ivan Chajda (Přerov)

MSC:

06A06	Partial orders, general
06A11	Algebraic aspects of posets

Keywords:

atomistic poset; statisch pair; finite element