Semiprime ideals and separation theorems for posets
type: Item
instance of: scholarly article
title: Semiprime ideals and separation theorems for posets
zbMATH Open document ID: 1155.06003
author: V. S. Kharat
author: Khalid A. Mokbel
published in: Order
publication date: 2008-11-17T00:00:00Z
review text: Let \(P\) be a poset and let \(A\) be a subset of \(P\). Define \(A^{u}:=\{x\in P : x\geq a \text{ for every } a\in A\}\). Dually define \(A^{l}:=\{x\in P : x\leq a \text{ for every } a\in A\}\). Then \(A^{ul}\) means \(\{A^{u}\}^l\) and \(A^{lu}\) means \(\{A^{l}\}^u\). A subset \(I\) of \(P\) is called an ideal if \(a,b\in I\) implies that \(\{a,b\}^{ul}\subseteq I\). A proper ideal \(I\) is called prime if \(\{a,b\}^{l}\subseteq I\) implies that either \(a\in I\) or \(b\in I\). Dually define filter and prime filter. These notions are due, respectively, to \textit{R. Halaš} [``Characterization of distributive sets by generalized annihilators'', Arch. Math., Brno 30, No.~1, 25--27 (1994; Zbl 0805.06001)] and \textit{R. Halaš} and \textit{J. Rachůnek} [``Polars and prime ideals in ordered sets'', Discuss. Math., Algebra Stoch. Methods 15, No. 1, 43--59 (1995; Zbl 0840.06003)]. In [``The theory of representations for Boolean algebras'', Trans. Am. Math. Soc. 40, 37--111 (1936; Zbl 0014.34002 and JFM 62.0033.04)], \textit{M. H. Stone} proved a separation theorem in the case of distributive lattices: Let \(L\) be a distributive lattice, let \(I\) be an ideal, let \(D\) be a filter of \(L\) such that \(I\cap D=\emptyset\). Then there exists a prime ideal \(P\) of \(L\) such that \(I\subseteq P\) and \(P\cap D=\emptyset\). In [``Semiprime ideals in general lattices'', J. Pure Appl. Algebra 56, No. 2, 105--118 (1989; Zbl 0665.06006)], \textit{Y. Rav} proved an analogue of Stone's Theorem for semiprime ideals in lattices. An ideal \(I\) of a lattice \(L\) is semiprime if \(x\wedge y\in I\) and \(x\wedge z\in I\) together imply \(x\wedge(y\vee z)\). In the paper under review the authors generalize the notion of semiprime ideal in lattices to posets as follows. An ideal \(I\) of a poset \(P\) is semiprime if \(\{a,b\}^{l}\subseteq I\) and \(\{a,c\}^{l}\subseteq I\) together imply \(\{a,\{b,c\}^{u}\}^{l}\subseteq I\). The authors then prove theorems analoguous to Stone's and …
reviewed by: Imed Zaguia
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 5366784
zbMATH Keywords: prime ideals
zbMATH Keywords: semiprime ideals
zbMATH Keywords: distributive posets
MaRDI profile type: Publication
full work available at URL: s11083-008-9087-3
OpenAlex ID: W2008281728
Wikidata QID: Q37060168
cites work: Ideals in Partially Ordered Sets
cites work: Q4399900
cites work: Q4306064
cites work: Primeness and semiprimeness in posets
cites work: Q3035342
cites work: Partially Ordered Sets
cites work: Semiprime ideals in general lattices
cites work: The Theory of Representation for Boolean Algebras
DBLP publication ID: journals/order/KharatM08
DOI: 10.1007/S11083-008-9087-3
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description: scientific article; zbMATH DE number 5366784