Semigroup

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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 x⋆y=y⋆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,⋆) to (T,∘) is a map from S to T satisfying

f(x⋆y)=f(x)∘f(y).

## Contents

* 1 Examples
* 2 Congruences
* 3 Cancellation property
* 4 Free semigroup

## Examples[edit]

* 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[edit]

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

a∼b and c∼d⇒a⋆c∼b⋆d.

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

[x]∘[y]=[x⋆y]

and this defines the quotient semigroup S/∼.

## Cancellation property[edit]

A semigroup satisfies the cancellation property if

xz=yz⇒x=y and
zx=zy⇒x=y.

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

## Free semigroup[edit]

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

n↔gn=gg⋯g.

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