Normal semilattices
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 \(a\wedge b=0\).
scientific article; zbMATH DE number 1283807