Normal semilattices
type: Item
instance of: scholarly article
title: Normal semilattices
zbMATH Open document ID: 0921.06003
author: K. Appert
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1999-06-20T00:00:00Z
review text: Throughout this paper \(S\) denotes a bounded \(\Lambda\)-semilattice with least element 0 and greatest element 1. For \(a,b\in S\), the relative annihilator of \(a\) in \(b\) is \(\langle a,b\rangle= \{x\in S\mid x\wedge a\leq b\}\). An ideal of \(S\) is a nonempty subset \(I\) of \(S\) such that: i) \(y\leq x\) and \(x\in I\Rightarrow y\in I\), and ii) for \(x,y\in I\) there exists \(z\in I\) such that \(z\geq x\) and \(z\geq y\). The authors give a new proof of a result of \textit{J. C. Varlet} [Bull. Aust. Math. Soc. 9, 169-185 (1973; Zbl 0258.06009)]: Theorem 1. \(S\) is distributive if and only if \(\langle a,b\rangle\) is an ideal for any \(a,b\in S\). A filter \(F\) of \(S\) is a non-empty subset of \(S\) such that \(x\wedge y\in F\iff x\in F\) and \(y\in F\). A proper filter \(P\) of \(S\) is prime if for any two filters \(F\) and \(K\) of \(S\), \(\emptyset\neq F\cap K\subseteq P\Rightarrow F\subseteq P\) or \(K\subseteq P\). A proper filter \(M\) in \(S\) is maximal if the only filter containing \(M\) is \(S\). For two ideals \(I\) and \(J\) of \(S\), \(I\vee J\) denotes the ideal generated by \(I\cup J\). A normal semilattice is a bounded distributive \(\Lambda\)-semilattice in which every prime filter is contained in a unique maximal filter. The main theorem is: Theorem 2. The following are equivalent for a bounded distributive semilattice \(S\): 1) \(S\) is normal; 2) \(P\cap \langle a,b\rangle\neq \emptyset\) or \(P\cap\langle b,a\rangle\neq \emptyset\) for any prime filter \(P\) of \(S\) and any \(a,b\in S\) and \(a\wedge b=0\); 3) \(\langle a,b\rangle\vee \langle b,a\rangle=S\) for any \(a,b\in S\) with \(a\wedge b=0\).
Mathematics Subject Classification ID: 06A12
zbMATH DE Number: 1283807
zbMATH Keywords: Boolean lattice
zbMATH Keywords: relative annihilator
zbMATH Keywords: ideal
zbMATH Keywords: filter
zbMATH Keywords: normal semilattice
zbMATH Keywords: bounded distributive semilattice
reviewed by: Thomas B. Muenzenberger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1283807