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Decompositions for relatively normal lattices. (English) Zbl 0799.06019

A lower-bounded distributive lattice is called relatively normal if in its set of prime ideals \(P\) ordered by set-inclusion every principal upper set is a chain. The most general conditions are obtained under which a relatively normal lattice may be represented as a union of its special ideals (Theorem B). It is also shown that if for a lower-bounded distributive lattice \(L\) its quotient lattice \(L/\theta\) relative to the Glivenko congruence \(\theta\) satisfies the descending chain condition, then \(L\) is relatively normal iff \(L\) is isomorphic to the lattice of all principal convex \(\ell\)-subgroups of an abelian \(\ell\)-group (Theorem D).

MSC:

06D05 Structure and representation theory of distributive lattices
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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