Definition 2.5.1.  Let M be a set, ∗ : M × M → M be an operation on M, e ∈ M. We define:
(M, ∗, e) forms a monoid  :⇔  (M, ∗) forms a semigroup and e is an identity of [M, ∗]  ⇔  ∗ is associative and e is an identity for ∗
Equivalence.  No proof.
References.