Semigroup: Definition, Examples, Properties

A semigroup in mathematics is a set equipped with a binary operation that is associative. In this article, we will study the definition of semigroups together with examples, and properties.

Definition of a Semigroup

Let G be a non-empty set and o be an algebraic operation acting on it. Then the pair (G, o) is called a semigroup if the following are satisfied:

  • G is closed under o. In other words, aob ∈ G for all a, b ∈ G.
  • o is associative, that is, (aob)oc = ao(boc) for all a,b,c ∈ G.

Examples of Semigroup

(R, +) is a semigroup where R is the set of real numbers.


1For any two real numbers a, b we have that a+b ∈ R. Thus, R is closed under +.
2As (a+b)+c=a+(b+c) for any real numbers a,b,c we have that + is associative on R.

Conclusion: Thus (R, +) is a semigroup and it follows from the above definition.

Non-Examples of Semigroups

(Z, -) is not a semigroup where Z denotes the set of all integers. This is because `-` is not associative on Z. See that (1-1)-2 ≠ 1-(1-2).

Properties of Semigroups

The list of properties satisfied by semigroups is given below.

  1. If a semigroup (G, o) is commutative, that is, aob=boa for all a,b ∈ G, then it is called a commutative semigroup or an abelian semigroup (named after the mathematician N. Abel). For example, (Z,+), (Q,+), (R,+) all are commutative semigroups.
  2. Let (G, o) be a semigroup and a ∈ G. Then am+n = amoan for all natural numbers m,n. This is because o is associative on G.
  3. A semigroup is called a finite order semigroup if it contains a finite number of elements. For example, (Zn, .) is a semigroup of order n.

Related Topics:

Group Theory: Definition, Examples, Orders, Types, Properties, Applications

Abelian Group: Definition, Properties, Examples

Cyclic Group: Definition, Orders, Properties, Examples

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