A term used as an abbreviation for
the phrase
"semi-group with identity" .
Thus, a
monoid
is a set
with an associative binary operation, usually called
multiplication,
in which there is an element
such that
for any
.
The element
is called the
identity
(or
unit)
and is usually denoted by
.
In any monoid there is exactly one identity. If
the operation given on the monoid is commutative, it is often called
addition
and the identity is called the
zero
and is denoted by
.
Examples of monoids.
1) The set of all mappings of an arbitrary set
into itself is a monoid relative to the operation
of successive application (composition) of mappings. The identity mapping is
the identity. 2) The set of endomorphisms of a
universal algebra
is a monoid relative to composition; the identity is the identity endomorphism. 3) Every
group
is a monoid.
Every semi-group
without an identity can be imbedded in a monoid. For this it suffices to take a symbol
not in
and give a multiplication on the set
as follows:
,
for any
,
and on elements from
the operation is as before. Every monoid can be represented
as the monoid of all endomorphisms of some universal algebra.
An arbitrary monoid can also be considered as a
category
with one object. This allows one to associate with a monoid
its dual (opposite, adjoint) monoid
.
The elements of both monoids coincide, but the product of
and
in
is put equal to the product
in
.
The development of the theory of monoids and adjoint functors has
shown the utility of the definition of a monoid in so-called
monoidal categories.
Suppose given a category
equipped with a bifunctor
,
an object
and natural isomorphisms
satisfying coherence conditions. An object
is called a monoid in the category
if there are morphisms
and
such that the following diagrams are commutative:
If
is taken to be the category of sets (cf.
Sets, category of),
the
Cartesian product,
a one-point set, and the isomorphisms
,
and
are chosen in the natural way
(
,
),
then the second definition of a monoid turns
out to be equivalent to the original definition.