# 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 $\star$ satisfying the following conditions:

• S is closed under $\star$;
• The operation $\star$ is associative.

A commutative semigroup is one which satisfies the further property that $x \star y = y \star x$ 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 $(S,{\star})$ to $(T,{\circ})$ is a map from S to T satisfying $f(x \star y) = f(x) \circ f(y) . \,$

## 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 $\sim\,$ which respects the binary operation: $( a \sim b \hbox{ and } c \sim d ) \Rightarrow ( a \star c \sim b \star d ) . \,$

The equivalence classes under a congruence can be given a semigroup structure $[x] \circ [y] = [x \star y] \,$

and this defines the quotient semigroup $S/\sim\,$.

## Cancellation property

A semigroup satisfies the cancellation property if $xz = yz \quad\Rightarrow\quad x = y, \,$ $zx = zy \quad\Rightarrow\quad x = y . \,$

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 (that is, 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 $n \leftrightarrow g^n = gg \cdots g . \,$

Every semigroup may be expressed as a quotient of a free semigroup. Some content on this page may previously have appeared on Citizendium.