commutative monoid
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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a monoid is a set equipped with an
associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
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, a binary operation ...
and an
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
. For example, the nonnegative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s with addition form a monoid, the identity element being . Monoids are
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s with identity. Such
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
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. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, 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, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
and
computer programming Computer programming or coding is the composition of sequences of instructions, called computer program, programs, that computers can follow to perform tasks. It involves designing and implementing algorithms, step-by-step specifications of proc ...
, 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 ...
. Transition monoids and
syntactic monoid In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the minimal monoid that recognizes the language L. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism. Syntactic quot ...
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 monoids and history monoids provide a foundation for process calculi 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 Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...
, the study of monoids is fundamental for automata theory (
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 correspon ...
), and
formal language theory In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a set called "#Definition, alphabet". The alphabet of a formal language consists of symbol ...
( star height problem). See
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
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 ...
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, a binary operation ...
, which we will denote , is a monoid if it satisfies the following two axioms: ; Associativity: For all , and in , the equation holds. ; Identity element: There exists an element in such that for every element in , the equalities and hold. In other words, a monoid is a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
with an
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
. 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 (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...
with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a constant, i. e. -ary (or nullary) operation. The monoid therefore is characterized by specification of the triple . Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by
juxtaposition Juxtaposition is an act or instance of placing two opposing elements close together or side by side. This is often done in order to Comparison, compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary ...
; 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, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
of that is closed under the monoid operation and contains the identity element of . Symbolically, is a submonoid of if , and whenever . In this case, is a monoid under the binary operation inherited from . On the other hand, if is a subset of a monoid that is closed under the monoid operation, and is a monoid for this inherited operation, then 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 a ...
is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the
nonnegative integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
s.


Generators

A subset of is said to ''generate'' if the smallest submonoid of containing is . If there is a finite set that generates , then 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. Perhaps most familiar as a pr ...
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 relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...
ing , defined by if there exists such that . An ''order-unit'' of a commutative monoid is an element of such that for any element of , there exists in the set generated by such that . This is often used in case 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 ...
, in which case we say that is an order-unit of .


Partially commutative monoid

A monoid for which the operation is commutative for some, but not all elements is a trace monoid; 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 while XOR and OR have the identity . 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s is a commutative monoid under addition (identity element ) or multiplication (identity element ). A submonoid of under addition is called a numerical monoid. * The set of
positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
s is a commutative monoid under multiplication (identity element ). * 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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
). * Generalizing the previous example, every bounded
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
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 a ...
closed under a binary operation forms the trivial (one-element) monoid, which is also the trivial group. * 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 In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
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 .) ** The
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 (for example, The set of all ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 for ...
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. For a vector, \vec\!, adding two matrices would have the geometric effect of applying each matrix transformation separately ...
or
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
as the operation. * The set of all finite strings over some fixed alphabet forms a monoid with string concatenation as the operation. The
empty string In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), characters such as letters, digits ...
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 ...
'' 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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
, with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, ). * 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. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
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 (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
. * 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 po ...
consisting of all
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
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, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
. 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, unc ...
. 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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
and an object of . The set of all endomorphisms of , denoted , forms a monoid under composition of
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s. For more on the relationship between category theory and monoids see below. * The set of
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
classes of compact surfaces with the connected sum. Its unit element is the class of the ordinary 2-sphere. Furthermore, if denotes the class of the
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
, and denotes the class of the projective plane, then every element of the monoid has a unique expression in the form where is a positive integer and , or . We have . * Let be a cyclic monoid of order , that is, . Then for some . 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 is then given by function composition. When then the function is a permutation of , and gives the unique
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
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 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.


Types of monoids

An inverse monoid is a monoid where for every in , there exists a unique in 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 be a monoid, with the binary operation denoted by and the identity element denoted by . Then a (left) -act (or left act over ) is a set together with an operation which is compatible with the monoid structure as follows: * for all in : ; * for all , in and in : . This is the analogue in monoid theory of a (left)
group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
. Right -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 tra ...
can be made into an operator monoid by adjoining the identity transformation.


Monoid homomorphisms

A
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between two monoids and is a function such that * for all , in * , where and are the identities on and respectively. Monoid homomorphisms are sometimes simply called monoid morphisms. Not every
semigroup homomorphism In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily t ...
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 the homomorphism. For example, consider , the set of
residue class In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mod ...
es modulo equipped with multiplication. In particular, is the identity element. Function given by is a semigroup homomorphism, since . However, , 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) whe ...
, as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only element such that ). A
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
monoid homomorphism is called a monoid
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
. 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 In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
. One does this by specifying a set of generators , 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 ...
. One does this by extending (finite)
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
s on 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 , which is then a monoid congruence. In the typical situation, the relation is simply given as a set of equations, so that . 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 (it has infinite order). Elements of this plactic monoid may be written as a^ib^j(ba)^k for integers , , , as the relations show that commutes with both and .


Relation to category theory

Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 . 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 Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 The ...
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 which is an abstract definition of what is a monoid in a category. A monoid object in Set 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, defined by its behavior (semantics) from the point of view of a '' user'' of the data, specifically in terms of possible values, possible operations on data ...
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 cal ...
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 or similar algorithm, in order to utilize multiple cores or processors efficiently. Given a sequence of values of type with identity element and associative operation , the ''fold'' operation is defined as follows: \mathrm: M^ \rarr M = \ell \mapsto \begin \varepsilon & \text \ell = \mathrm \\ m \bullet \mathrm \, \ell' & \text \ell = \mathrm \, m \, \ell' \end In addition, any
data structure In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships amo ...
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 tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...
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), tree data stru ...
.


MapReduce

An application of monoids in computer science is the so-called MapReduce 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 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 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 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 consists ...
such that \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'. An ordered commutative monoid is a commutative monoid together with a partial ordering 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(a + S) 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.


See also

* Cartesian 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) *
Semiring In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
and Kleene algebra * Star height problem * Vedic square *
Frobenioid In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of ...


Notes


Citations


References

* * * * * * * * * * * * *


External links

* * * {{PlanetMath, urlname=Monoid , title=Monoid , id=389 Algebraic structures Category theory Semigroup theory