Almost principal element lattices (Q1895282)

A \(C\)-lattice is a multiplicative lattice \(L\) every element of which is the join of elements of a multiplicatively closed subset \(C\) of compact elements. Let \(m\) be a prime element of \(L\); denote by \(L_m\) the localization of \(L\) at the filter \(\{a\in C\); \(a\nleq m\}\) on \(C\). If \(L_m\) is a principal element lattice for any maximal element \(m\in L\), then \(L\) is called the almost principal element lattice. The authors present the following equivalent conditions under which a \(C\)-lattice \(L\) is an almost principal element lattice: (1) \(L\) is locally noetherian and every compact element of \(L\) is principal. (2) \(L\) is locally noetherian and for any maximal element \(m\in L\) the interval \([m^2, m]\) is totally ordered. (3) \(L\) is locally noetherian and for every maximal element \(m\in L\) the interval \([m^2, m]\) is simple or trivial. (4) \(L\) is locally noetherian and for every maximal element \(m\in L\) every \(m\)-primary element is a power of \(m\). (5) \(L\) is locally noetherian, distributive and satisfies the weak union condition. Here, the weak union condition means: if \(a,b,c\in L\), \(a\nleq b\) and \(a\nleq c\) then there exists a principal element \(d\leq a\) with \(d\nleq b\) and \(d\nleq c\).