The alternating group A_n is a submonoid of the symmetric group S_n. For, the product of two even permutations is again even and the identity map is even.

In fact, it is also a submonoid of all S_m for m > n.

Both S_n and A_n are submonoids of the monoid of all maps of {1, ..., n} to itself.

The set of elements of R[X] which take the value 0 at some fixed element a form a submonoid of (R[X],+,0).

The set of elements of R[X] which take the value 1 at some fixed element a form a submonoid of (R[X],*,1).

The matrices in M_n(R) with determinant 1 form a submonoid, denoted by SL_n(R), of the monoid defined on M_n(R) by matrix multiplication. Indeed, if A, B SL_n(R), then

det(A) det(B) = det(AB) = 1.

Moreover, the identity matrix also has determinant 1.

A second submonoid of M_n(R) is formed by the set of matrices with determinant not equal to 0. This submonoid is denoted by GL_n(R). Notice that  SL_n(R)  is also a submonoid of GL_n(R).

The set of even integers  2Z is closed under addition and multiplication. The even integers form a submonoid of Z with respect to addition, but not with respect to multiplication as 1 is not even.