Modular Pairs, Covering Property and Related Results in Posets

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Authors

  • N. K. Thakare
     13, General Arun Kumar Vaidya Nagar, Off Sakri Road, Dhule-424001 ,IN
  • S. Maeda
     Iwaidani 6-333-10, Matsuyama 790 ,JP
  • B. N. Waphare
     Department of Mathematics, University of Pune, Pune-411007 ,IN

Keywords:

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

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.

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Published

2003-12-01

How to Cite

Thakare, N. K., Maeda, S., & Waphare, B. N. (2003). Modular Pairs, Covering Property and Related Results in Posets. The Journal of the Indian Mathematical Society, 70(1-4), 229–253. Retrieved from https://www.informaticsjournals.com/index.php/jims/article/view/21984

Issue

Volume 70, Issue 1-4, January-December 2003

Section

Articles

License

Copyright (c) 2003 N. K. Thakare, S. Maeda, B. N. Waphare

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.