= Chain union of partial group operations
{title2=$m=\bigcup\mathcal C$}
For a nonempty inclusion chain of <group> operations on <subsets> of a fixed <set>, the union is a <group operation> on the union of their carriers. Any finite list of carrier elements lies in one chain member, so its multiplication and <associativity> agree there. Nested identities coincide because the only <idempotent> of a <group> is its identity; inverses remain valid in larger members. Thus the union supplies the chain <upper bound> required by <Zorn's lemma>.
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