Prime ideals in 0-distributive posets

Vinayak Joshi; Nilesh Mundlik

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 940-955
  • ISSN: 2391-5455

Abstract

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In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.

How to cite

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Vinayak Joshi, and Nilesh Mundlik. "Prime ideals in 0-distributive posets." Open Mathematics 11.5 (2013): 940-955. <http://eudml.org/doc/269405>.

@article{VinayakJoshi2013,
abstract = {In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.},
author = {Vinayak Joshi, Nilesh Mundlik},
journal = {Open Mathematics},
keywords = {0-distributive poset; Prime ideal; Minimal prime ideal; Dense ideal; Maximal l-filter; Pseudocomplemented poset; prime ideal; minimal prime ideal; dense ideal; maximal 1-filter; pseudocomplemented poset},
language = {eng},
number = {5},
pages = {940-955},
title = {Prime ideals in 0-distributive posets},
url = {http://eudml.org/doc/269405},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Vinayak Joshi
AU - Nilesh Mundlik
TI - Prime ideals in 0-distributive posets
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 940
EP - 955
AB - In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.
LA - eng
KW - 0-distributive poset; Prime ideal; Minimal prime ideal; Dense ideal; Maximal l-filter; Pseudocomplemented poset; prime ideal; minimal prime ideal; dense ideal; maximal 1-filter; pseudocomplemented poset
UR - http://eudml.org/doc/269405
ER -

References

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