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For a certain family of complete modular lattices, we prove aJordan--H\"older--Scheier-like" theorem with no assumptions on cardinality orwell-orderedness. This family includes both lattices which are both join- andmeet-continuous, as well as the lattices of subobjects of any object in anabelian category satisfying properties related to Grothendieck's axioms (AB5)and (AB5*). We then give several examples of objects in abelian categorieswhich satisfy these axioms, including pointwise finite-dimensional persistencemodules, presheaves, and certain Pr\"ufer modules. Moreover, we show that, overan arbitrary ring, the infinite product of isomorphic simple modules both failsto satisfy our axioms and admits at least two composition series with distinctcardinalities. We conclude by giving a lattice-theoretic proof that any objectwhich is locally finitely generated and satisfies our axioms can be expressedas a direct sum of indecomposable subobjects. We conjecture that thisdecomposition is unique.

Composition series of arbitrary cardinality in modular lattices and abelian categories