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2013 | 11 | 5 | 940-955

Tytuł artykułu

Prime ideals in 0-distributive posets

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EN

Abstrakty

EN
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.

Wydawca

Czasopismo

Rocznik

Tom

11

Numer

5

Strony

940-955

Opis fizyczny

Daty

wydano
2013-05-01
online
2013-03-14

Twórcy

  • University of Pune
  • Nowrosjee Wadia College of Arts and Science

Bibliografia

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  • [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
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  • [5] Erné M., Verallgemeinerungen der Verbandstheorie II: m-Ideale in halbgeordneten Mengen und Hüllenräumen, Habilitationsschrift, University of Hannover, 1979
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  • [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
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  • [11] Frink O., Ideals in partially ordered sets, Amer. Math. Monthly, 1954, 61, 223–234 http://dx.doi.org/10.2307/2306387
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  • [13] Grätzer G., General Lattice Theory, 2nd ed., Birkhäuser, Basel, 1998
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  • [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
  • [17] Halaš R., Rachůnek J., Polars and prime ideals in ordered sets, Discuss. Math. Algebra Stochastic Methods, 1995, 15(1), 43–59
  • [18] Joshi V., On completion of section semicomplemented posets, Southeast Asian Bull. Math., 2007, 31(5), 881–892
  • [19] Joshi V.V., Waphare B.N., Characterizations of 0-distributive posets, Math. Bohem., 2005, 130(1), 73–80
  • [20] Kaplansky I., Commutative Rings, University of Chicago Press, Chicago, 1974
  • [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
  • [22] Kharat V.S., Mokbel K.A., Primeness and semiprimeness in posets, Math. Bohem., 2009, 134(1), 19–30
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  • [24] Mokbel K.A., A study of ideals and central elements in partially ordered sets, PhD thesis, University of Pune, 2007
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  • [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
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  • [30] Stone M.H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc., 1936, 40(1), 37–111
  • [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
  • [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
  • [33] Varlet J.C., A generalization of the notion of pseudo-complementedness, Bull. Soc. Roy. Sci. Liège, 1968, 36, 149–158
  • [34] Venkatanarasimhan P.V., Semi-ideals in posets, Math. Ann., 1970, 185(4), 338–348 http://dx.doi.org/10.1007/BF01349957
  • [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
  • [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
  • [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
  • [38] Waphare B.N., Joshi V., On distributive pairs in posets, Southeast Asian Bull. Math., 2007, 31(6), 1205–1233

Typ dokumentu

Bibliografia

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bwmeta1.element.doi-10_2478_s11533-013-0206-z
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