# Semigroup

In algebra, a **semigroup** is a set equipped with a binary operation satisfying certain properties similar to but less stringent than those of a group. A motivating example of a semigroup is the set of positive integers with multiplication as the operation.

Formally, a semigroup is a set *S* with a binary operation satisfying the following conditions:

*S*is closed under ;- The operation is associative.

A *commutative semigroup* is one which satisfies the further property that for all *x* and *y* in *S*. Commutative semigroups are often written additively.

A *subsemigroup* of *S* is a subset *T* of *S* which is closed under the binary operation and hence is again a semigroup.

A semigroup *homomorphism* *f* from semigroup to is a map from *S* to *T* satisfying

## Examples

- The positive integers under addition form a commutative semigroup.
- The positive integers under multiplication form a commutative semigroup.
- Square matrices under matrix multiplication form a semigroup, not in general commutative.
- Every monoid is a semigroup, by "forgetting" the identity element.
- Every group is a semigroup, by "forgetting" the identity element and inverse operation.

## Congruences

A **congruence** on a semigroup *S* is an equivalence relation which respects the binary operation:

The equivalence classes under a congruence can be given a semigroup structure

and this defines the **quotient semigroup** .

## Cancellation property

A semigroup satisfies the *cancellation property* if

- and

A semigroup is a subsemigroup of a group if and only if it satisfies the cancellation property.

## Free semigroup

The **free semigroup** on a set *G* of *generators* is the set of all words on *G*, the finite sequences of elements of *G*, with the binary operation being concatenation (juxtaposition). The free semigroup on one generator *g* may be identified with the semigroup of positive integers under addition

Every semigroup may be expressed as a quotient of a free semigroup.