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.