Commutative binary operation
From Groupprops
This article defines a property of binary operations (and hence, of magmas)
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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 × 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.
