commutative
Let S be a set and ∘ a binary operation on it. ∘ is said to be commutative
if
a∘b=b∘a |
for all a,b∈S.
Viewing ∘ as a function from S×S to S, the commutativity of ∘ can be notated as
∘(a,b)=∘(b,a). |
Some common examples of commutative operations are
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addition over the integers: m+n=m+n for all integers m,n
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multiplication over the integers: m⋅n=m⋅n for all integers m,n
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addition over n×n matrices, A+B=B+A for all n×n matrices A,B, and
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multiplication over the reals: rs=sr, for all real numbers r,s.
A binary operation that is not commutative is said to be non-commutative. A common example of a non-commutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,
a-b≠b-a. |
For instance, 2-1=1≠-1=1-2.
Other examples of non-commutative binary operations can be found in the attachment below.
Remark. The notion of commutativity can be generalized to n-ary operations, where n≥2. An n-ary operation f on a set A is said to be commutative if
f(a1,a2,…,an)=f(aπ(1),aπ(2),…,aπ(n)) |
for every permutation π on {1,2,…,n}, and for every choice of n elements ai of A.
Title | commutative |
Canonical name | Commutative |
Date of creation | 2013-03-22 12:22:45 |
Last modified on | 2013-03-22 12:22:45 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 20-00 |
Synonym | commutativity |
Synonym | commutative law |
Related topic | Associative |
Related topic | AbelianGroup2 |
Related topic | QuantumTopos |
Related topic | NonCommutativeStructureAndOperation |
Related topic | Subcommutative |
Defines | non-commutative |