Modular Pairs, Standard Elements, Neutral Elements and Related Results in Partially Ordered Sets

Abstract

In an attempt to answer the open problem of Birkhoff, namely, "How should one define modular pairs in a general poset?”, five distinct notions of modular pair in a general poset are discussed and studied. Interrelationships between these five concepts are looked into and several counter-examples are supplied. The covering properties and the exchange property in general posets are thoroughly studied and several characterizations of these properties are obtained. As an offshoot of our study, twentythree characterizations of the covering property in atomistic lattices are accomplished.

Investigations of del-relation, del-tilda relation, perspectivity, sub-perspectivity etc. in posets are carried out.

In the context of another open problem posed by Birkhoff concerning how to define natural extensions of the concepts of neutral elements and standard elements we cover substantial ground and supply adequate solution to this query. Special posets such as SSC-poseLs, SSC*-posets, orthomodular and orthocomplemented posets are studied and several characterizations are obtained.