The Journal of the Indian Mathematical Society

N. K. Thakare ¹, M. M. Pawar ², B. N. Waphare ³

Affiliations
1 4B, Mitrakunj Society, Sakri Road, Dhule-424 001, IN
2 Department of Mathematics, S.S.V.P.S’s. L.K. Dr. PR. Ghogrey Science College, Dhule-424 005, IN
3 Department of Mathematics, University of Pune, Pune 411 007, IN

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.

Keywords

Modular Pair, Covering Property, Exchange Property, Del-Relation, Del-Tilda Relation, Perspectivity, Subperspectivity, g-Atomic Poset, g-Atomistic Poset, SSC-Poset, SSC*-Poset, Standard Element, Neutral Element, Central Element, g-Central Element.

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N. K. Thakare ¹, S. Maeda ², B. N. Waphare ³

Affiliations
1 13, General Arun Kumar Vaidya Nagar, Off Sakri Road, Dhule-424001, IN
2 Iwaidani 6-333-10, Matsuyama 790, JP
3 Department of Mathematics, University of Pune, Pune-411007, IN

Abstract

How should one define a modular pair in a general poset? queried Birkhoff, sixty years ago. Not only we answer this open problem satisfactorily but obtain interesting properties concerning covering property, exchange property in a poset with zero. In this context a few counter examples are also supplied. This general study has led us, as an offshoot, to thirteen characterizations of covering property in a lattice with zero. Del-relation and perspectivity are characterized in posets. The study of atom spaces is extended to posets. Statischness in atomistic posets is characterized. Further, ortho-modular posets are also characterized and an interesting open problem in this context is raised.

Keywords

Poset, Upper (Lower) Cone, Semilattice, Lattice, Covering Relation, Atom, Modular Pair, Covering Property, Exchange Property, Del-Relation, Perspectivity, Atom Space, Orthomodular Posets.

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