A monoid that is commutative i.e., a monoid such that for every two elements and in , . This means that commutative monoids are commutative, associative, and have an identity element.
For example, the nonnegative integers under addition form a commutative monoid. The integers under the operation with all form a commutative monoid. This monoid collapses to a group only if and are restricted to the integers 0, 1, ..., , since only then do the elements have unique additive inverses. Similarly, the integers under the operation also forms a commutative monoid.
The numbers of commutative monoids of orders , 2, ... are 1, 2, 5, 19, 78, 421, 2637, ... (OEIS A058131).