Commutative binary operation

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

Definition

Definition with symbols

Let $S$ be a set and $*$ be a binary operation on $S$ (viz, $*$ is a map $S$ × $S$ → $S$. 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.