Let be a semigroup and let be a nonempty subset. Recall that the subsemigroup of generated by is the set . (Since is associative, the notation is unambiguous provided we insist that .) Let denote the set of all subsemigroups of which contain . (Note that is nonempty since it contains .) Prove that .
Note that if is a subsemigroup containing , then in particular we have for all and . (We can see this by using induction on .) So , and thus .
Note also that is itself a subsemigroup of , by generalized associativity. That is, if , then where for and for . So .