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The hull-kernel topology on prime ideals in posets

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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.

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Acknowledgments

The authors are grateful to the referees for their critical and valuable suggestions. Also, the authors thank Professor B. N. Waphare for his fruitful comments.

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Authors and Affiliations

  1. Department of Mathematics, Nowrosjee Wadia College of Arts and Science, Pune, 411001, India

    Nilesh Mundlik

  2. Department of Mathematics, Savitribai Phule Pune University, Pune, 411007, India

    Vinayak Joshi

  3. Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, Olomouc, Czech Republic

    Radomír Halaš

Authors
  1. Nilesh Mundlik
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  2. Vinayak Joshi
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  3. Radomír Halaš
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Corresponding author

Correspondence to Vinayak Joshi.

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The authors declare that there is no conflict of interests regarding the publishing of this paper.

Additional information

Communicated by A. Di Nola.

The work of the third author is supported by the international project Austrian Science Fund (FWF)-Grant Agency of the Czech Republic (GAČR) 15-346971L, by the AKTION project “Ordered structures for Algebraic Logic” 71p3 and by the Palacký University project IGA PrF 2015010.

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Mundlik, N., Joshi, V. & Halaš, R. The hull-kernel topology on prime ideals in posets. Soft Comput 21, 1653–1665 (2017). https://doi.org/10.1007/s00500-016-2105-2

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