Main Index
Algebraic structures
Algebraic operations
Subject Index
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Main Index
Algebraic structures
Algebraic operations
Subject Index
comment on the page
A binary operation on a set 
is a mapping 
which associates every ordered pair 
of elements of 
with a unique element 
. The multiplicative notation 
used for the image element carries no meaning except what is given by its definition. 1
A binary operation 
on a set 
is commutative (or abelian) if it satisfies the commutative law: 
for all 
and 
in 
. Otherwise, the operation is non-commutative.
A binary operation 
on a set 
is called associative if it satisfies the associative law: 
for all 
and 
in 
. Therefore when the operation 
is associative, the evaluation order can be left unspecified without causing ambiguity.
An external binary operation is a binary function from 
to 
. This differs from a binary operation defined above in that 
need not be 
; its elements come from outside. A well known example of such external binary operation is the scalar multiplication in linear algebra. Here 
is a field and 
is a vector space over that field.
| 1 | The used notation  is called the infix notation  . Other types of notations are the prefix notation  , or postfix notation  . We can also simply write  . |
Cite this web-page as:
Štefan Porubský: Binary Operation.