Commutative binary operation
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 be a set and
be a binary operation on
(viz,
is a map
×
→
. Then,
is said to be commutative if, for every
in
, the following identity holds:
If the above equation holds for particular values of and
, we say that
and
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.