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**Almost principal element lattices.**
*(English)*
Zbl 0829.06013

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 \(d\nleq c\).

(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 \(d\nleq c\).

Reviewer: V.Novák (Brno)

### MSC:

06F10 | Noether lattices |