Semigroup

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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 ${\displaystyle \star }$ satisfying the following conditions:

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

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

${\displaystyle f(x\star y)=f(x)\circ f(y).\,}$

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 ${\displaystyle \sim \,}$ which respects the binary operation:

${\displaystyle a\sim b{\hbox{ and }}c\sim d\Rightarrow a\star c\sim b\star d.\,}$

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

${\displaystyle [x]\circ [y]=[x\star y]\,}$

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

Cancellation property

A semigroup satisfies the cancellation property if

${\displaystyle xz=yz\Rightarrow x=y\,}$ and
${\displaystyle zx=zy\Rightarrow x=y.\,}$

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, 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

${\displaystyle n\leftrightarrow g^{n}=gg\cdots g.\,}$

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