The hull-kernel topology on prime ideals in posets

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, Erne and Joshi (Discrete Math 338:954---971, 2015). We study the hull-kernel topology on the set of all prime ideals $$\mathcal {P}(Q)$$P(Q), minimal prime ideals $$\mathrm{Min}(Q)$$Min(Q) and maximal ideals $$\mathrm{Max}(Q)$$Max(Q) of a poset Q. Then topological properties like compactness, connectedness and separation axioms of $$\mathcal {P}(Q)$$P(Q) are studied. Further, we focus on the space of minimal prime ideals $$\mathrm{Min}(Q)$$Min(Q) of a poset Q. Under the additional assumption that every maximal ideal is prime, the collection of all maximal ideals $$\mathrm{Max}(Q)$$Max(Q) of a poset Q forms a subspace of $$\mathcal {P}(Q)$$P(Q). Finally, we prove a characterization of a space of maximal ideals of a poset to be a normal space.