A commutative monoid is a fundamental structure in mathematics. It is a set of elements with an associative binary operation that has an identity element. The operation is also commutative, which means that the order of the elements doesn't matter. Synonyms for this term include Abelian monoid, commutative semigroup, and commutative groupoid. The Abelian monoid emphasizes the commutativity of the operation, while the commutative semigroup emphasizes the associativity and commutativity. A commutative groupoid is a more general version of a commutative monoid, where the operation may not have an identity element. Other related concepts include commutative rings, commutative algebra, and commutative geometry.