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- W4386648112 abstract "For a certain family of complete modular lattices, we prove a Jordan--Holder--Scheier-like theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5*). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prufer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique." @default.
- W4386648112 created "2023-09-13" @default.
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- W4386648112 date "2023-11-01" @default.
- W4386648112 modified "2023-10-09" @default.
- W4386648112 title "Composition series of arbitrary cardinality in modular lattices and abelian categories" @default.
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- W4386648112 doi "https://doi.org/10.1016/j.aim.2023.109292" @default.
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