Maximal and completely meet irreducible sublattices of a distributive lattice
For a lattice L we use the following notations: \({\mathcal P}(L)\) resp. \({\mathcal Q}(L)\) for the set of all prime ideals resp. filters of L and (x] resp. [x) the ideal resp. filter generated by x. A lattice is distributive iff it contains sufficiently many prime ideals (enough to separate every two elemnts); Stone (1936); Iseki (1951). Using prime filters too and extending a work of Hashimoto (1952) the author proves a separation property for a pair: element-sublattice: Lemma 2.1. - Let S be a sublattice of a distributive lattice. For \(x\in L\backslash S\), each of the following holds: (i) If \(S\cap [x)=\emptyset\) then there exists \(I\in {\mathcal P}(L)\) such that \(x\not\in I\) and \(S\subseteq I\); (ii) If \(S\cap(x]=\emptyset\) then there exists \(F\in {\mathcal Q}(L)\) such that \(x\not\in F\) and \(S\subseteq F\); (iii) If \(S\cap [x)\neq \emptyset\) and \(S\cap(x]\neq \emptyset\) then there exist \(I\in {\mathcal P}(L)\) and \(F\in {\mathcal Q}(L)\) such that \(x\not\in I\cup F\) and \(S\subseteq I\cup F\). - From this lemma the main result follows easily. Theorem 2.2. - Let L be a distributive lattice. A subset \(\emptyset \neq S\subseteq L\) is a proper sublattice of L iff there exist index sets \(J_ 1,J_ 2,J_ 3\) such that \[ S=(\cap_{j\in J_ 1}I_ j)\cap(\cap_{j\in J_ 2}F_ j)\cap(\cap_{j\in J_ 3}(I_ j\cup F_ j)) \] where \(I_ j\in {\mathcal P}(L)\), \(F_ j\in {\mathcal Q}(L)\), \(I_ j\cup F_ j\neq L\) for all \(j\in J_ 3\) and \(J_ 1\cup J_ 2\cup J_ 3\neq \emptyset.\) This characterization is very prolific. The author proves characterizations for maximal and, more generally, for completely meet irreducible sublattices and applies them for finite distributive lattices, Boolean algebras, free distributive lattices and Post algebras of order \(n\geq 2\), recovering results of Rival, Koh, Chen and Tan concerning maximal sublattices. We mention Theorem 4.2. - Let S be a sublattice of a distributive lattice L. Then S is completely meet irreducible iff one of the following holds; (1) \(S=L\); (2) S is a maximal filter and L has a 0; (3) S is a maximal ideal and L has a 1; (4) \(S=I\cup F\) where (i) I is a completely meet irreducible ideal of L, (ii) F is a completely meet irreducible filter of L, \((iii)\quad(\cap \{G\in {\mathcal Q}(L)| \quad F\subset G\})\cup(\cap \{J\in {\mathcal P}(L)| \quad I\subset J\})\neq \emptyset.\)
scientific article; zbMATH DE number 3853159