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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The authors declare that there is no conflict of interests regarding the publishing of this paper.
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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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DOI: https://doi.org/10.1007/s00500-016-2105-2