A monoid that is commutative i.e., a monoid M such that for every two elements a and b in M, a b = b a. 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 mod(x + y, n) with n element Z^+ all form a commutative monoid. This monoid collapses to a group only if x and y are restricted to the integers 0, 1, ..., n - 1, since only then do the elements have unique additive inverses. Similarly, the integers under the operation max(x + y, n) also forms a commutative monoid.

monoid | semigroup | submonoid