Prime ideals in 0-distributive posets

Vinayak Joshi; Nilesh Mundlik

Open Mathematics (2013)

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

Abstract

top
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

top

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

top
  1. [1] Balasubramani P., Characterizations of the 0-distributive semilattice, Math. Bohem., 2003, 128(3), 237–252 Zbl1052.06002
  2. [2] Batueva C., Semenova M., Ideals in distributive posets, Cent. Eur. J. Math., 2011, 9(6), 1380–1388 http://dx.doi.org/10.2478/s11533-011-0075-2 Zbl1242.06002
  3. [3] Chajda I., Complemented ordered sets, Arch. Math. (Brno), 1992, 28(1–2), 25–34 Zbl0785.06002
  4. [4] David E., Erné M., Ideal completion and Stone representation of ideal-distributive ordered sets, Topology Appl., 1992, 44(1–3), 95–113 http://dx.doi.org/10.1016/0166-8641(92)90083-C Zbl0768.06003
  5. [5] Erné M., Verallgemeinerungen der Verbandstheorie II: m-Ideale in halbgeordneten Mengen und Hüllenräumen, Habilitationsschrift, University of Hannover, 1979 
  6. [6] Erné M., Distributivgesetze und Dedekind’sche Schnitte, Abh. Braunschweig. Wiss. Ges., 1982, 33, 117–145 Zbl0526.06005
  7. [7] Erné M., Distributive laws for concept lattices, Algebra Universalis, 1993, 30(4), 538–580 http://dx.doi.org/10.1007/BF01195382 Zbl0795.06006
  8. [8] Erné M., Prime ideal theory for general algebras, Appl. Categ. Structures, 2000, 8(1–2), 115–144 http://dx.doi.org/10.1023/A:1008611926427 
  9. [9] Erné M., Prime and maximal ideals of partially ordered sets, Math. Slovaca, 2006, 56(1), 1–22 Zbl1164.03011
  10. [10] Erné M., Wilke G., Standard completions for quasiordered sets, Semigroup Forum, 1983, 27(1–4), 351–376 http://dx.doi.org/10.1007/BF02572747 Zbl0517.06009
  11. [11] Frink O., Ideals in partially ordered sets, Amer. Math. Monthly, 1954, 61, 223–234 http://dx.doi.org/10.2307/2306387 Zbl0055.25901
  12. [12] Gorbunov V.A., Tumanov V.I., On the existence of prime ideals in semidistributive lattices, Algebra Universalis, 1983, 16, 250–252 http://dx.doi.org/10.1007/BF01191774 Zbl0516.06006
  13. [13] Grätzer G., General Lattice Theory, 2nd ed., Birkhäuser, Basel, 1998 Zbl0909.06002
  14. [14] Halaš R., Annihilators and ideals in ordered sets, Czechoslovak Math. J., 1995, 45(120)(1), 127–134 Zbl0838.06003
  15. [15] Halaš R., Some properties of Boolean ordered sets, Czechoslovak Math. J., 1996, 46(121)(1), 93–98 Zbl0904.06002
  16. [16] Halaš R., Joshi V., Kharat V.S., On n-normal posets, Cent. Eur. J. Math., 2010, 8(5), 985–991 http://dx.doi.org/10.2478/s11533-010-0062-z Zbl1234.06003
  17. [17] Halaš R., Rachůnek J., Polars and prime ideals in ordered sets, Discuss. Math. Algebra Stochastic Methods, 1995, 15(1), 43–59 Zbl0840.06003
  18. [18] Joshi V., On completion of section semicomplemented posets, Southeast Asian Bull. Math., 2007, 31(5), 881–892 Zbl1150.06001
  19. [19] Joshi V.V., Waphare B.N., Characterizations of 0-distributive posets, Math. Bohem., 2005, 130(1), 73–80 Zbl1112.06001
  20. [20] Kaplansky I., Commutative Rings, University of Chicago Press, Chicago, 1974 
  21. [21] Kharat V.S., Mokbel K.A., Semiprime ideals and separation theorems for posets, Order, 2008, 25(3), 195–210 http://dx.doi.org/10.1007/s11083-008-9087-3 Zbl1155.06003
  22. [22] Kharat V.S., Mokbel K.A., Primeness and semiprimeness in posets, Math. Bohem., 2009, 134(1), 19–30 Zbl1212.06001
  23. [23] Larmerová J., Rachůnek J., Translations of distributive and modular ordered sets, Acta. Univ. Palack. Olomuc. Fac. Rerum. Natur. Math., 1988, 27, 13–23 Zbl0693.06003
  24. [24] Mokbel K.A., A study of ideals and central elements in partially ordered sets, PhD thesis, University of Pune, 2007 
  25. [25] Nachbin L., Une proprietété caractéristique des algèbres booléiennes, Portugal. Math., 1947, 6, 115–118 Zbl0034.16603
  26. [26] Niederle J., Boolean and distributive ordered sets: Characterization and representation by sets, Order, 1995, 12(2), 189–210 http://dx.doi.org/10.1007/BF01108627 Zbl0838.06004
  27. [27] Pawar Y.S., Lokhande A.D., 0–1-distributivity and complementedness, Bull. Calcutta Math. Soc., 1998, 90(2), 147–150 
  28. [28] Pawar Y.S., Thakare N.K., 0-distributive semilattices, Canad. Math. Bull., 1978, 21(4), 469–481 http://dx.doi.org/10.4153/CMB-1978-080-6 Zbl0413.06002
  29. [29] Rav Y., Semiprime ideals in general lattices, J. Pure Appl. Algebra, 1989, 56(2), 105–118 http://dx.doi.org/10.1016/0022-4049(89)90140-0 
  30. [30] Stone M.H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc., 1936, 40(1), 37–111 Zbl0014.34002
  31. [31] Thakare N.K., Pawar M.M., Waphare B.N., Modular pairs, standard elements, neutral elements and related results in partially ordered sets, J. Indian Math. Soc. (N.S.), 2004, 71(1–4), 13–53 Zbl1117.06301
  32. [32] Thakare N.K., Pawar Y.S., Minimal prime ideals in 0-distributive semilattices, Period. Math. Hungar., 1982, 13(3), 237–246 http://dx.doi.org/10.1007/BF01847920 Zbl0532.06003
  33. [33] Varlet J.C., A generalization of the notion of pseudo-complementedness, Bull. Soc. Roy. Sci. Liège, 1968, 36, 149–158 Zbl0162.03501
  34. [34] Venkatanarasimhan P.V., Semi-ideals in posets, Math. Ann., 1970, 185(4), 338–348 http://dx.doi.org/10.1007/BF01349957 Zbl0182.33902
  35. [35] Venkatanarasimhan P.V., Pseudo-complements in posets, Proc. Amer. Math. Soc., 1971, 28(1), 9–17 http://dx.doi.org/10.1090/S0002-9939-1971-0272687-X Zbl0218.06002
  36. [36] Waphare B.N., Joshi V., Characterization of standard elements in posets, Order, 2004, 21(1), 49–60 http://dx.doi.org/10.1007/s11083-004-2862-x Zbl1060.06003
  37. [37] Waphare B.N., Joshi V.V., On uniquely complemented posets, Order, 2005, 22(1), 11–20 http://dx.doi.org/10.1007/s11083-005-9002-0 Zbl1091.06002
  38. [38] Waphare B.N., Joshi V., On distributive pairs in posets, Southeast Asian Bull. Math., 2007, 31(6), 1205–1233 Zbl1150.06006

NotesEmbed ?

top

You must be logged in to post comments.