Prime ideals in 0-distributive posets

Abstract

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

Ideals in atomic posets

“…For the case of 3-ideals, some of the results in this paper are to be found in [31]. The m-ideals always form an ideal completion, the m-ideal completion I m Q .…”

“…Notice that in case m > 1, every subset I satisfying the implication B ⊂ m I ⇒ B ⊆ I is automatically a down-set. The following special cases are included in our general definition: the 0-ideals are the down-sets, alias order ideals or semi-ideals [31], the 1-ideals and the 2-ideals are the down-sets containing Q ℓ , the 3-ideals are the ideals in [30][31][32]34] containing Q ℓ , the ω-ideals are the ideals in the sense of Frink [21] (cf. [2,8,17,39]), the ω 1 -ideals (for the least uncountable cardinal ω 1 ) are the σ -ideals [35], the Ω-ideals are the cuts, alias closed ideals [3] or normal ideals [25].…”

“…Now, let us consider a few local and global variants of the usual distributive laws in lattices, and their generalizations to posets (see e.g. [8,11,20,30,31,36,50]). Our first definition generalizes the classical notion of distributivity (for elements) in lattices: an element a of a poset Q is called m-distributive if…”

“…For posets instead of lattices, the discussion becomes more subtle, because there are various reasonable notions of ideals, of distributivity, of primeness, etc., and the absence of joins or meets may cause serious problems. This area of order theory was investigated in a series of papers by Erné [10][11][12][13][14][15][16][17][18][19][20] and independently by Chajda, Halas, Larmerová, Rachůnek, Niederle [5,6,26,27,29,30,36,39], and later by Joshi, Kharat, Mokbel, Mundlik, Waphare [31,32,34,49,50] and many others.…”

The hull-kernel topology on prime ideals in posets

“…During last few years, the theory of prime ideals of posets has been developed; see David and Erné (1992), Erné (2006), Erné and Joshi (2015), Halaš et al (2010), Halaš and Rachůnek (1995) and, Joshi and Mundlik (2013). In fact, Venkatanarasimhan (1970) seems to be first who introduced the topology on the set of all prime semi-ideals of a poset and characterized T 1 -spaces.…”