Soft Computing

, Volume 21, Issue 7, pp 1653–1665 | Cite as

The hull-kernel topology on prime ideals in posets

Foundations

Abstract

In this paper, we continue our study of prime ideals in posets that was started in Joshi and Mundlik (Cent Eur J Math 11(5):940–955, 2013) and, Erné and Joshi (Discrete Math 338:954–971, 2015). We study the hull-kernel topology on the set of all prime ideals \(\mathcal {P}(Q)\), minimal prime ideals \(\mathrm{Min}(Q)\) and maximal ideals \(\mathrm{Max}(Q)\) of a poset Q. Then topological properties like compactness, connectedness and separation axioms of \(\mathcal {P}(Q)\) are studied. Further, we focus on the space of minimal prime ideals \(\mathrm{Min}(Q)\) of a poset Q. Under the additional assumption that every maximal ideal is prime, the collection of all maximal ideals \(\mathrm{Max}(Q)\) of a poset Q forms a subspace of \(\mathcal {P}(Q)\). Finally, we prove a characterization of a space of maximal ideals of a poset to be a normal space.

Keywords

Prime ideal Minimal prime ideal Maximal ideal  Pseudocomplemented poset Quasicomplemented poset 0-Distributive poset 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Nilesh Mundlik
    • 1
  • Vinayak Joshi
    • 2
  • Radomír Halaš
    • 3
  1. 1.Department of MathematicsNowrosjee Wadia College of Arts and SciencePuneIndia
  2. 2.Department of MathematicsSavitribai Phule Pune UniversityPuneIndia
  3. 3.Department of Algebra and Geometry, Faculty of SciencePalacký University OlomoucOlomoucCzech Republic

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