With respect to both addition and multiplication, these sets are semi-groups having a unit, so they are monoids. We call them the additive and multiplicative monoids on N, Z, Q, R, C, Z/nZ, R[X], ..., respectively.

There are two natural ways to make the set M_n(R) of all real n × n-matrices a monoid:

• The monoid multiplication is matrix multiplication, the unit element is the identity matrix.
• The monoid multiplication is matrix addition. The unit element is the zero matrix (all entries of the matrix are equal to 0).

The symmetric group S_n with composition as binary operation and the identity as unit, is a monoid.

In computer science, one often writes A^* for the set of all words over the alphabet A. This is the monoid with respect to concatenation, whose unit is the `empty word', the word consisting of 0 letters.