Commutative Monoid
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).
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