Definition 2.5.8.  Let M be a set, :M × M → M be an operation on M, e ∈ M. We define:
(M, , e) forms a commutative monoid  :⇔  (M, , e) forms a monoid and [M, , e] is commutative  ⇔   is associative and  is commutative and e is an identity for 
Equivalence.  No proof.