A commutative monoid is a monoid 
whose operation is commutative,
so 
for every two elements 
and 
in 
. As a monoid, it has an associative operation and an identity
element.
For example, the nonnegative integers under addition form a commutative monoid. For a positive
integer 
,
the residue representatives 0, 1, ..., 
under the operation 
form an Abelian group,
and hence a commutative monoid. For a nonnegative
integer 
,
another example is the integers 0, 1, ..., 
under capped addition 
, whose identity
is 0.
The numbers of commutative monoids with 
, 2, ... elements, counted up
to isomorphism, are 1, 2, 5, 19, 78, 421, 2637, ...
(OEIS A058131).
