Main Index
Algebraic structures
Structures with one operation
Groupoids
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Main Index
Algebraic structures
Structures with one operation
Groupoids
Subject Index
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A monoid is a triple 
where 
is a non-empty set and 
is a binary operation such that
is an associative operation, that is 
for every 
, and
is an identity element with respect to 
, i.e. 
for every 
. 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
for all 
, and
.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.