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Some remarks on modular QFD lattices. (English) Zbl 1150.06009

The aim of the present paper is to prove for lattices some results which come from module theory. The main result of the paper is Proposition 1.6, a characterization of upper continuous modular lattices which are compactly generated and QFD (i.e., every interval \([x,1]\) has no infinite independent subsets). This result is applied to Grothendieck categories in Proposition 3.11. Another interesting result is Proposition 2.10, which provides a sufficient condition for the existence of dual Krull dimension for the poset of all subdirectly irreducible elements of an interval \([a,b]\) in an almost compactly generated modular lattice.

MSC:

06C05 Modular lattices, Desarguesian lattices
06B35 Continuous lattices and posets, applications
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18E15 Grothendieck categories (MSC2010)
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