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Algebraic operations
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The notion of the operation (especially that of the binary operation) is the keystone of the algebra.
Let be a non-empty set and
a non-negative integer.1 Let
,
times for
and
. Under an
-ary operation (or shortly an operation) on
we understand a function
. This means that an
-ary operation assigns to every ordered
-tuple of elements of
a unique element from
.
The number is called the arity or the rank of the operation
and the inputs of the underlying function are called operands.
Sometimes it is useful to consider a constant as an operation of arity 0, such operation is called nullary. Thus
A non-empty subset of
is called closed with respect to the operation
defined on
if the restriction
is an operation on
.
Binary relation on a set
is called compatible with an
-ary operation
defined on
if
for all
such that
,
.
A non-empty set endowed with one or more operations is called algebraic structure or algebra. More precisely, an (universal) algebra is a pair
, in which
is a non-empty set, and
is a set operations on
.
Let and
be two non-empty sets endowed with
-ary operations
and
, respectively, where
is a non-negative integer. A map
is called a homomorphism (with respect to operations
and
) if
In words, if maps the result under the operation
to the result of
-maps under the operation
. If
is one-to-one (injective) and onto (surjective), that is,
is a bijection, then
is called an isomorphism.
Lemma: Let and
be two non-empty sets endowed with
-ary operations
and
, respectively, where
is a non-negative integer and map
a bijection. If
is a homomorphism, then so its inverse
.
Let , is a system of non-empty sets
each with an
-ary operation
, then we can define (the so-called componentwise definition) a new operation
on the Cartesian product
by
In many cases it is useful to drop the assumption that is a function with a domain
, that is that
is defined for ordered
-tuples of elements of
. If
is defined only on a subset of
then
is called a partial
-ary operation. Typical example of a partial binary operation is the division of real numbers where division by
is not defined.
1 | In general ![]() |
Cite this web-page as:
Štefan Porubský: Algebraic Operation.