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 \star satisfying the following conditions:

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) . \,

Contents

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 \Rightarrow x = y \, and
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

n \leftrightarrow g^n = gg \cdots g . \,

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

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