TheInfoList

OR: In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a monoid is a set equipped with an
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
and an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. For example, the nonnegative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
and
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, the set of strings built from a given set of characters is a
free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ele ...
.
Transition monoid In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' ...
s and
syntactic monoid In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L. Syntactic quotient The free monoid on a given set is the monoid whose elements are all the strings of z ...
s are used in describing
finite-state machine A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
s.
Trace monoid In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing cert ...
s and history monoids provide a foundation for
process calculi In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions, communications, and ...
and
concurrent computing Concurrent computing is a form of computing in which several computations are executed '' concurrently''—during overlapping time periods—instead of ''sequentially—''with one completing before the next starts. This is a property of a syst ...
. In
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the t ...
, the study of monoids is fundamental for
automata theory Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματ� ...
(
Krohn–Rhodes theory In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond t ...
), and
formal language theory In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of s ...
(
star height problem The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. Specifically, is a nesting depth ...
). See semigroup for the history of the subject, and some other general properties of monoids.

# Definition

A
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'' equipped with a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
, which we will denote •, is a monoid if it satisfies the following two axioms: ; Associativity: For all ''a'', ''b'' and ''c'' in ''S'', the equation holds. ; Identity element: There exists an element ''e'' in ''S'' such that for every element ''a'' in ''S'', the equalities and hold. In other words, a monoid is a semigroup with an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. It can also be thought of as a
magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural ...
with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a constant, i. e. 0-ary (or nullary) operation. The monoid therefore is characterized by specification of the triple (''S'', • , ''e''). Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written and . This notation does not imply that it is numbers being multiplied. A monoid in which each element has an inverse is a group.

# Monoid structures

## Submonoids

A submonoid of a monoid is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''N'' of ''M'' that is closed under the monoid operation and contains the identity element ''e'' of ''M''. Symbolically, ''N'' is a submonoid of ''M'' if , whenever , and . In this case, ''N'' is a monoid under the binary operation inherited from ''M''. On the other hand, if ''N'' is subset of a monoid that is closed under the monoid operation, and is a monoid for this inherited operation, then ''N'' is not always a submonoid, since the identity elements may differ. For example, the
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the
nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s.

## Generators

A subset ''S'' of ''M'' is said to ''generate'' ''M'' if the smallest submonoid of ''M'' containing ''S'' is ''M''. If there is a finite set that generates ''M'', then ''M'' is said to be a finitely generated monoid.

## Commutative monoid

A monoid whose operation is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
is called a commutative monoid (or, less commonly, an abelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with its ''algebraic''
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...
ing , defined by if there exists ''z'' such that . An ''order-unit'' of a commutative monoid ''M'' is an element ''u'' of ''M'' such that for any element ''x'' of ''M'', there exists ''v'' in the set generated by ''u'' such that . This is often used in case ''M'' is the positive cone of a partially ordered
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
''G'', in which case we say that ''u'' is an order-unit of ''G''.

## Partially commutative monoid

A monoid for which the operation is commutative for some, but not all elements is a
trace monoid In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing cert ...
; trace monoids commonly occur in the theory of concurrent computation.

# Examples

* Out of the 16 possible binary Boolean operators, four have a two-sided identity that is also commutative and associative. These four each make the set a commutative monoid. Under the standard definitions, AND and XNOR have the identity True while XOR and OR have the identity False. The monoids from AND and OR are also
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
while those from XOR and XNOR are not. * The set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s $\N = \$ is a commutative monoid under addition (identity element 0) or multiplication (identity element 1). A submonoid of under addition is called a numerical monoid. * The set of
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s $\N \setminus \$ is a commutative monoid under multiplication (identity element 1). * Given a set , the set of subsets of is a commutative monoid under intersection (identity element is itself). * Given a set , the set of subsets of is a commutative monoid under union (identity element is the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
). * Generalizing the previous example, every bounded semilattice is an
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
commutative monoid. ** In particular, any bounded lattice can be endowed with both a meet- and a join- monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices, Heyting algebras and
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
s are endowed with these monoid structures. * Every
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
closed under a binary operation • forms the trivial (one-element) monoid, which is also the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
. * Every group is a monoid and every
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
a commutative monoid. * Any semigroup may be turned into a monoid simply by adjoining an element not in and defining for all . This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids. ** Thus, an idempotent monoid (sometimes known as ''find-first'') may be formed by adjoining an identity element to the left zero semigroup over a set . The opposite monoid (sometimes called ''find-last'') is formed from the right zero semigroup over . *** Adjoin an identity to the left-zero semigroup with two elements . Then the resulting idempotent monoid models the
lexicographical order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
of a sequence given the orders of its elements, with ''e'' representing equality. * The underlying set of any ring, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity 1.) ** The
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, with addition or multiplication as operation. ** The set of all by matrices over a given ring, with
matrix addition In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Krone ...
or matrix multiplication as the operation. * The set of all finite strings over some fixed alphabet forms a monoid with
string concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenati ...
as the operation. The
empty string In formal language theory, the empty string, or empty word, is the unique string of length zero. Formal theory Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special c ...
serves as the identity element. This monoid is denoted and is called the ''
free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ele ...
'' over . It is not commutative if has at least two elements. * Given any monoid , the ''opposite monoid'' has the same carrier set and identity element as , and its operation is defined by . Any commutative monoid is the opposite monoid of itself. * Given two sets and endowed with monoid structure (or, in general, any finite number of monoids, , their
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
is also a monoid (respectively, ). The associative operation and the identity element are defined pairwise. * Fix a monoid . The set of all functions from a given set to is also a monoid. The identity element is a
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
mapping any value to the identity of ; the associative operation is defined
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
. * Fix a monoid with the operation and identity element , and consider its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is pos ...
consisting of all
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of . A binary operation for such subsets can be defined by . This turns into a monoid with identity element . In the same way the power set of a group is a monoid under the product of group subsets. * Let be a set. The set of all functions forms a monoid under
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. The identity is just the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
. It is also called the '' full transformation monoid'' of . If is finite with elements, the monoid of functions on is finite with elements. * Generalizing the previous example, let be a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
and an object of . The set of all
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gro ...
s of , denoted , forms a monoid under composition of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s. For more on the relationship between category theory and monoids see below. * The set of
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
classes of compact surfaces with the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...
. Its unit element is the class of the ordinary 2-sphere. Furthermore, if denotes the class of the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
, and ''b'' denotes the class of the projective plane, then every element ''c'' of the monoid has a unique expression the form where is a positive integer and , or . We have . * Let $\langle f\rangle$ be a cyclic monoid of order , that is, $\langle f\rangle = \left\$. Then $f^n = f^k$ for some $0 \le k < n$. In fact, each such gives a distinct monoid of order , and every cyclic monoid is isomorphic to one of these.
Moreover, can be considered as a function on the points $\$ given by $\begin 0 & 1 & 2 & \cdots & n-2 & n-1 \\ 1 & 2 & 3 & \cdots & n-1 & k\end$ or, equivalently $f(i) := \begin i+1, & \text 0 \le i < n-1 \\ k, & \text i = n-1. \end$ Multiplication of elements in $\langle f\rangle$ is then given by function composition. When $k = 0$ then the function is a permutation of $\,$ and gives the unique
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
of order .

# Properties

The monoid axioms imply that the identity element is unique: If and are identity elements of a monoid, then .

## Products and powers

For each nonnegative integer , one can define the product $p_n = \textstyle \prod_^n a_i$ of any sequence $\left(a_1,\ldots,a_n\right)$ of elements of a monoid recursively: let and let for . As a special case, one can define nonnegative integer powers of an element of a monoid: and for . Then for all .

## Invertible elements

An element is called
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
if there exists an element such that and . The element is called the inverse of . Inverses, if they exist, are unique: If and are inverses of , then by associativity . If is invertible, say with inverse , then one can define negative powers of by setting for each ; this makes the equation hold for all . The set of all invertible elements in a monoid, together with the operation •, forms a group.

## Grothendieck group

Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements and exist such that holds even though is not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse of would get that , which is not true. A monoid has the cancellation property (or is cancellative) if for all , and in , the equality implies , and the equality implies . A commutative monoid with the cancellation property can always be embedded in a group via the ''Grothendieck group construction''. That is how the additive group of the integers (a group with operation +) is constructed from the additive monoid of natural numbers (a commutative monoid with operation + and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group. If a monoid has the cancellation property and is ''finite'', then it is in fact a group. The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group. The cancellative property in a monoid is not necessary to perform the Grothendieck construction – commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, if , then and have the same image in the Grothendieck group, even if . In particular, if the monoid has an absorbing element, then its Grothendieck group is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
.

## Types of monoids

An inverse monoid is a monoid where for every ''a'' in ''M'', there exists a unique ''a''−1 in ''M'' such that and . If an inverse monoid is cancellative, then it is a group. In the opposite direction, a '' zerosumfree monoid'' is an additively written monoid in which implies that and : equivalently, that no element other than zero has an additive inverse.

# Acts and operator monoids

Let ''M'' be a monoid, with the binary operation denoted by • and the identity element denoted by ''e''. Then a (left) ''M''-act (or left act over ''M'') is a set ''X'' together with an operation which is compatible with the monoid structure as follows: * for all ''x'' in ''X'': ; * for all ''a'', ''b'' in ''M'' and ''x'' in ''X'': . This is the analogue in monoid theory of a (left)
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
. Right ''M''-acts are defined in a similar way. A monoid with an act is also known as an '' operator monoid''. Important examples include transition systems of semiautomata. A
transformation semigroup In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transf ...
can be made into an operator monoid by adjoining the identity transformation.

# Monoid homomorphisms A
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
between two monoids and is a function such that * for all ''x'', ''y'' in ''M'' * , where ''e''''M'' and ''e''''N'' are the identities on ''M'' and ''N'' respectively. Monoid homomorphisms are sometimes simply called monoid morphisms. Not every semigroup homomorphism between monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of homomorphism. For example, consider $M_n$, the set of
residue class In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
es modulo $n$ equipped with multiplication. In particular, the class of $1$ is the identity. Function $f\colon M_3\to M_6$ given by $f\left(k\right)=3k$ is a semigroup homomorphism as $3k\cdot 3l = 9kl = 3kl$ in $M_6$. However, $f\left(1\right)=3 \neq 1$, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted. In contrast, a semigroup homomorphism between groups is always a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
, as it necessarily preserves the identity (because, in a group, the identity is the only element such that ). A
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
monoid homomorphism is called a monoid
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.

# Equational presentation

Monoids may be given a ''presentation'', much in the same way that groups can be specified by means of a group presentation. One does this by specifying a set of generators Σ, and a set of relations on the
free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ele ...
Σ. One does this by extending (finite)
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s on Σ to monoid congruences, and then constructing the quotient monoid, as above. Given a binary relation , one defines its symmetric closure as . This can be extended to a symmetric relation by defining if and only if and for some strings with . Finally, one takes the reflexive and transitive closure of ''E'', which is then a monoid congruence. In the typical situation, the relation ''R'' is simply given as a set of equations, so that $R=\$. Thus, for example, : $\langle p,q\,\vert\; pq=1\rangle$ is the equational presentation for the bicyclic monoid, and : $\langle a,b \,\vert\; aba=baa, bba=bab\rangle$ is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as $a^ib^j\left(ba\right)^k$ for integers ''i'', ''j'', ''k'', as the relations show that ''ba'' commutes with both ''a'' and ''b''.

# Relation to category theory

Monoids can be viewed as a special class of
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
. Indeed, the axioms required of a monoid operation are exactly those required of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
composition when restricted to the set of all morphisms whose source and target is a given object. That is, : ''A monoid is, essentially, the same thing as a category with a single object.'' More precisely, given a monoid , one can construct a small category with only one object and whose morphisms are the elements of ''M''. The composition of morphisms is given by the monoid operation •. Likewise, monoid homomorphisms are just
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s between single object categories. So this construction gives an equivalence between the category of (small) monoids Mon and a full subcategory of the category of (small) categories Cat. Similarly, the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There ...
is equivalent to another full subcategory of Cat. In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. There is also a notion of
monoid object In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η ...
which is an abstract definition of what is a monoid in a category. A monoid object in
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is just a monoid.

# Monoids in computer science

In computer science, many
abstract data types In computer science, an abstract data type (ADT) is a mathematical model for data types. An abstract data type is defined by its behavior (semantics) from the point of view of a ''user'', of the data, specifically in terms of possible values, pos ...
can be endowed with a monoid structure. In a common pattern, a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of elements of a monoid is " folded" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be parallelized by employing a
prefix sum In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the sums of prefixes ( running totals) of the input sequence: : : : :... For instance, the prefix sums ...
or similar algorithm, in order to utilize multiple cores or processors efficiently. Given a sequence of values of type ''M'' with identity element $\varepsilon$ and associative operation $\bullet$, the ''fold'' operation is defined as follows: : $\mathrm: M^ \rarr M = \ell \mapsto \begin \varepsilon & \mbox \ell = \mathrm \\ m \bullet \mathrm \, \ell\text{'} & \mbox \ell = \mathrm \, m \, \ell\text{'} \end$ In addition, any
data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, a ...
can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a
binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
might differ depending on pre-order vs. post-order
tree traversal In computer science, tree traversal (also known as tree search and walking the tree) is a form of graph traversal and refers to the process of visiting (e.g. retrieving, updating, or deleting) each node in a tree data structure, exactly once. S ...
.

# MapReduce

An application of monoids in computer science is the so-called
MapReduce MapReduce is a programming model and an associated implementation for processing and generating big data sets with a parallel, distributed algorithm on a cluster. A MapReduce program is composed of a ''map'' procedure, which performs filterin ...
programming model (se
Encoding Map-Reduce As A Monoid With Left Folding
. MapReduce, in computing, consists of two or three operations. Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. "Reduce" consists of folding those elements, so that in the end we produce just one element. For example, if we have a
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case, the
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
is being sharded. To finalize reduction properly, the "Shuffling" stage regroups the data among the nodes. If we do not need this step, the whole Map/Reduce consists of mapping and reducing; both operations are parallelizable, the former due to its element-wise nature, the latter due to associativity of the monoid.

# Complete monoids

A complete monoid is a commutative monoid equipped with an
infinitary In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operat ...
sum operation $\Sigma_I$ for any
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consist ...
such thatDroste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. , pp. 7–10 : $\sum_ =0;\quad \sum_ = m_j;\quad \sum_ = m_j+m_k \quad \text j\neq k$ and : $\sum_ = \sum_ m_i \quad \text \bigcup_ I_j=I \text I_j \cap I_ = \emptyset \quad \text j\neq j\text{'}$. An ordered commutative monoid is a commutative monoid together with a
partial ordering In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
such that for every , and implies for all . A continuous monoid is an ordered commutative monoid in which every directed subset has a least upper bound, and these least upper bounds are compatible with the monoid operation: : $a + \sup S = \sup\left(a + S\right)$ for every and directed subset of . If is a continuous monoid, then for any index set and collection of elements , one can define : $\sum_I a_i = \sup_ \; \sum_E a_i,$ and together with this infinitary sum operation is a complete monoid.

*
Green's relations In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951 ...
*
Monad (functional programming) In functional programming, a monad is a software design pattern with a structure that combines program fragments ( functions) and wraps their return values in a type with additional computation. In addition to defining a wrapping monadic type, ...
*
Semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
and Kleene algebra *
Star height problem The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. Specifically, is a nesting depth ...
* Vedic square * Frobenioid

* * * * *