## 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 $$d\nleq c$$.
Reviewer: V.Novák (Brno)

### MSC:

 06F10 Noether lattices
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