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A monoid is a triple where
is a non-empty set and
is a binary operation such that
A monoid is also called a semigroup with an identity element.
Monoid is called commutative or Abelian if the binary operation is commutative, i.e.
for every
. Commutative monoids are often written additively.
Any commutative monoid can be endowed with an algebraic preordering <=, defined by x <= y if and only if there exists z such that x + z = y.
A submonoid of a monoid is a subset
such that
and
is also a monoid. In another words,
contains the identity element of
and
is closed under the operation
(that is, if
then
). Monoid
is also called overmonoid of
.
A monoid homomorphism (or simply homomorphism) between two monoids
and
is a mapping
such that
If is one-to-one then
is called a monoid isomorphism or simply isomorphism. If
is also onto, then
is called a monoid bijection (or shortly only bijection).
A monoid is called free if it contains a subset
possessing the property that every element of
can be uniquely written as a finite product of elements of
, that is there are elements
such that
and if
is another subset of elements of
with
then
and
for every
. Elements of the set
are called generators of
, and
is the generating set of
with respect to the operation
. If the monoid
has a finite set of generators its is called finitely generated.
Free monoids are those objects in the category of monoinds which satisfy the usual universal property defining free objects. Consequently, every monoid arises as a homomorphic image of a free monoid.
Each free monoid has exactly one set of free generators. The cardinality of the set of generators is called the rank of
. Two free monoids are isomorphic if and only if they have the same rank.
If there exists a subset of a monoid
such that every element of
can be represented uniquely up to the order as a product of elements of
then
is called free commutative with the set of generators
. For instance, the set of positive integers
is free commutative over the infinite set of prime numbers (fundamental theorem of arithmetic), nevertheless
is not a free monoid. But the set of natural numbers
under addition is a free monoid on a single generator, namely the number 1 (this the unique free generator for
.
Cite this web-page as:
Štefan Porubský: Monoid.