Commutative binary operation

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This article defines a property of binary operations (and hence, of magmas)


This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Definition with symbols

Let S be a set and * be a binary operation on S (viz, * is a map S × SS. Then, * is said to be commutative if, for every a, b, c in S, the following identity holds:

a * b = b * a

If the above equation holds for particular values of a and b, we say that a and b commute.

Related term

A magma where the binary operation is commutative is termed a commutative magma. For a semigroup, monoid or group, we use the word Abelian as an alternative to commutative (thus, a group where the binary operation is commutative is termed an Abelian group).

Related element properties

Central element

Further information: central element An element in a magma is termed central if it commutes with every element. The set of central elements of a magma is termed the commutative center.