Consistent dually semimodular lattices. (English) Zbl 0758.06005
Summary: We study combinatorial and order-theoretical properties of consistent dually semimodular lattices of finite rank. Natural examples of such lattices are lattices of subnormal subgroups of groups with composition series. We prove that a dually semimodular lattice is consistent if and only if it is “locally” atomic and modular. From this, we conclude that dually semimodular subgroup lattices are necessarily consistent. We discuss exchange properties for irredundant decompositions of elements into join-irreducibles. Finally, we show that a finite consistent dually semimodular lattice is modular if and only if it has the same number of join-irreducibles and meet-irreducibles.
Keywords:
modularity; consistent dually semimodular lattices; finite rank; lattices of subnormal subgroups of groups; exchange properties for irredundant decompositions of elementsReferences:
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