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Algebraic operations
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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 ![]() ![]() ![]() ![]() ![]() |
Cite this web-page as:
Štefan Porubský: Binary Operation.