In the first section of this paper, we prove an analogue of Stone's Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.
MSC:06B10, 06A12
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 P(Q), minimal prime ideals Min(Q) and maximal ideals Max(Q) of a poset Q. Then topological properties like compactness, connectedness and separation axioms of P(Q) are studied. Further, we focus on the space of minimal prime ideals Min(Q) of a poset Q. Under the additional assumption that every maximal ideal is prime, the collection of all maximal ideals Max(Q) of a poset Q forms a subspace of P(Q). Finally, we prove a characterization of a space of maximal ideals of a poset to be a normal space.
In this paper, we study Baer ideals in posets and obtain some characterizations of Baer ideals in [Formula: see text]-distributive posets. Further, we prove that in an ideal-distributive poset, every ideal is Baer (normal) if and only if every prime ideal is Baer (normal). We extend the concept of a quasicomplement to posets and prove characterizations of quasicomplemented poset. This extend the results regarding Baer ideals and quasicomplemented lattices mentioned in [Y. S. Pawar and S. S. Khopade, [Formula: see text]-ideals and annihilator ideals in 0-distributive lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49(1) (2010) 63–74; Y. S. Pawar and D. N. Mane, [Formula: see text]-ideals in 0-distributive semilattices and 0-distributive lattices, Indian J. Pure Appl. Math. 24(7–8) (1993) 435–443] to posets.
In this paper, we define the concepts of the radical of an ideal and a primary ideal in posets. Further, the analogue of the first and the second uniqueness theorems regarding primary decomposition of an ideal are obtained. In the last section, we prove that if an ideal in a poset Q has a minimal primary decomposition, then the diameter of the corresponding zero-divisor graph with respect to this ideal is exactly equal to three.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.