Mundlik, N., Joshi, V. & Halaš, R. Soft Comput (2017) 21: 1653. doi:10.1007/s00500-016-2105-2

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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.

Keywords

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

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Funding Note

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