Scarce decomposition of finite distributive lattices (Q2759838)

Let \((L_s: s\in S)\) be a system of finite lattices and let \(S\) be also a finite lattice. Moreover, suppose that the following three conditions are satisfied: (i) \(s\leq t\) and \(L_s\cap L_t\neq\emptyset\) imply that \(L_s\cap L_t\) is a filter of \(L_s\) and an ideal of \(L_t\); (ii) if \(t\) covers \(s\) in \(S\), then \(L_s\cap L_t\neq\emptyset\) and (iii) \(L_s\cap L_t\subseteq L_{s\wedge t} \cap L_{s\vee t}\). Set \(A= \bigcup(L_s:s\in S)\) and take the relation \(\leq\) on \(A\) which is the transitive hull of all partial orderings \(\leq_s\), \(s\in S\). Then, following \textit{C. Herrmann} [Math. Z. 130, 255-274 (1973; Zbl 0275.06007)], \((A;\leq)\) is a modular lattice whenever \(L_s\) are modular. Note that \textit{A. Wronski} [Rep. Math. Logic 2, 63-75 (1974; Zbl 0312.02024)] studied the \(S\)-sum construction for the case of distributive lattices \(L_s\), \(s\in S\).NEWLINENEWLINENEWLINEIt can be shown that every finite distributive lattice \(L\) can be written as an \(S\)-sum of some Boolean sublattices of \(L\). In general, such a decomposition of \(L\) need not be scarce. This means: \(L_s\nsubseteq L_t\) whenever \(s\neq t\). The author shows that a finite distributive lattice \(L\) possesses a scarce \(S\)-sum of some nontrivial sublattices of \(L\) if and only if every nontrivial convex sublattice of \(S\) contains a prime ideal.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].