an:00548942
Zbl 0799.06019
Hart, James B.; Tsinakis, Constantine
Decompositions for relatively normal lattices
EN
Trans. Am. Math. Soc. 341, No. 2, 519-548 (1994).
00018626
1994
j
06D05 06F20
root-system; filet configuration; distributive lattice; prime ideals; relatively normal lattice; quotient lattice; Glivenko congruence; descending chain condition; principal convex \(\ell\)-subgroups; abelian \(\ell\)-group
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).
V.N.Salij (Saratov)