Chain union of partial group operations ($m=\bigcup \mathcal{C}$)

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.