(Z,+), (Z, ·), (Z/nZ,+), (Z/nZ, ·), (R[X],+), (R[X], ·), (Q[X]/(f),+), (Q[X]/(f),·), ...

(2Z,+) and (2Z,·) are also semi-groups.

Let X be a set. The set of all maps X -> X is a semi-group with respect to composition of maps.

Let A be an alphabet, e.g., A={a,b, ..., z}. The set of all words over the alphabet A, that is, all strings of letters from A, is a semigroup with respect to concatenation, that is, the operation that just puts the second argument behind the first. For example, using . for the operation,

conca . tenation = concatenation