commutative
Let $S$ be a set and $\circ $ a binary operation^{} on it. $\circ $ is said to be commutative^{} if
$$a\circ b=b\circ a$$ 
for all $a,b\in S$.
Viewing $\circ $ as a function^{} from $S\times S$ to $S$, the commutativity of $\circ $ can be notated as
$$\circ (a,b)=\circ (b,a).$$ 
Some common examples of commutative operations are

•
addition over the integers: $m+n=m+n$ for all integers $m,n$

•
multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$

•
addition over $n\times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$, and

•
multiplication over the reals: $rs=sr$, for all real numbers $r,s$.
A binary operation that is not commutative is said to be noncommutative. A common example of a noncommutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,
$$ab\ne ba.$$ 
For instance, $21=1\ne 1=12$.
Other examples of noncommutative binary operations can be found in the attachment below.
Remark. The notion of commutativity can be generalized to $n$ary operations, where $n\ge 2$. An $n$ary operation $f$ on a set $A$ is said to be commutative if
$$f({a}_{1},{a}_{2},\mathrm{\dots},{a}_{n})=f({a}_{\pi (1)},{a}_{\pi (2)},\mathrm{\dots},{a}_{\pi (n)})$$ 
for every permutation $\pi $ on $\{1,2,\mathrm{\dots},n\}$, and for every choice of $n$ elements ${a}_{i}$ of $A$.
Title  commutative 
Canonical name  Commutative 
Date of creation  20130322 12:22:45 
Last modified on  20130322 12:22:45 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 2000 
Synonym  commutativity 
Synonym  commutative law 
Related topic  Associative 
Related topic  AbelianGroup2 
Related topic  QuantumTopos 
Related topic  NonCommutativeStructureAndOperation 
Related topic  Subcommutative 
Defines  noncommutative 