TheInfoList In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a monoid is a set equipped with an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Monoids are
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
s with identity. Such
algebraic structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s occur in several branches of mathematics. For example, the functions from a set into itself form a monoid with respect to function composition. More generally, in
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, the morphisms of an
object Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
and
computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...
, the set of
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
built from a given set of
characters Character(s) may refer to: Arts, entertainment, and media Literature * Character (novel), ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * Characters (Theophrastus), ''Characters'' (Theophrastus), a classical Greek set of cha ...
is a
free monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
.
Transition monoid In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s and
syntactic monoid In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
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 A model is an informative representation of an object, person or system. ...
s.
Trace monoid In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , an ...
s and
history monoidIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s provide a foundation for
process calculi In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of compu ...
and
concurrent computing Concurrent computing is a form of computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both compu ...
. In
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ... , the study of monoids is fundamental for
automata theory Automata theory is the study of abstract machines and automaton, automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' (the plural of ''automaton'') com ... (
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 theory, automata that seeks to decompose them in terms of elementary components. These compon ...
), and
formal language theory In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed ...
(
star height problem The star height problem in formal language theory is the question whether all regular languages can be expressed using Regular expression#Formal language theory, regular expressions of limited star height, i.e. with a limited nesting depth of Klee ...
). See
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
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) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formul ...
''S'' equipped with a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 equations and hold. In other words, a monoid is a
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
with an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. It can also be thought of as a
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
, i. e. 0-ary (or nullary) operation. The monoid therefore is characterized by specification of the
triple Triple is used in several contexts to mean "threefold" or a "Treble (disambiguation), treble": Sports * Triple (baseball), a three-base hit * A basketball three-point field goal * A figure skating jump with three rotations * In bowling terms, thre ...
(''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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
is a
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
.

# Monoid structures

## Submonoids

A submonoid of a monoid is a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... ''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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the
nonnegative integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 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 (mathematics), set. A poset consists of a set toget ...
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
''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 Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , an ...
; trace monoids commonly occur in the theory of
concurrent computation Concurrent computing is a form of computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both compu ...
.

# Examples

* Out of the 16 possible binary Boolean operators, each of the four that has a two-sided identity is also commutative and associative and thus makes the set a commutative monoid. Under the standard definitions,
AND And or AND may refer to: Logic, grammar, and computing * Conjunction (grammar) In grammar In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study o ...
and
XNOR The XNOR gate (sometimes ENOR, EXNOR or NXOR and pronounced as Exclusive NOR. Alternatively XAND, pronounced Exclusive AND) is a digital logic gate A logic gate is an idealized model of computation A model is an informative representation of ...
have the identity True while
XOR Exclusive or or exclusive disjunction is a logical operation In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic ...
and OR have the identity False. The monoids from AND and OR are also
idempotent Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... ). * Generalizing the previous example, every bounded
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilatti ...
is an
idempotent Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
commutative monoid. ** In particular, any bounded lattice can be endowed with both a
meet Meet may refer to: People with the name * Janek Meet Janek Meet (born 2 May 1974 in Viljandi) is a retired Estonians, Estonian football (soccer), footballer, who played in the Meistriliiga, for FC Kuressaare, whom he joined from JK Viljandi Tu ... - and a
join - monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices,
Heyting algebra __notoc__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch Dutch commonly refers to: * Something of, from, or related to the Netherlands * Dutch people () * Dutch language () *Dutch language , spoken in Belgium (also referred as ''fl ...
s and
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s are endowed with these monoid structures. * Every
singleton set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
closed under a binary operation • forms the trivial (one-element) monoid, which is also the
trivial group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. * Every
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
is a monoid and every
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
a commutative monoid. * Any
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
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 In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a sequence given the orders of its elements, with ''e'' representing equality. * The underlying set of any
ring Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity 1.) ** The
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s, with addition or multiplication as operation. ** The set of all by
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
over a given ring, with
matrix addition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
or
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... as the operation. * The set of all finite
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
over some fixed alphabet forms a monoid with
string concatenation In formal language theory In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and a ...
as the operation. The
empty string In formal language theory In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:� ...
serves as the identity element. This monoid is denoted and is called the ''
free monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
'' over . It is not commutative. * Given any monoid , the ''opposite monoid'' has the same carrier set and identity element as , and its operation is defined by . Any
commutative monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... mapping any value to the identity of ; the associative operation is defined
pointwise In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. * Fix a monoid with the operation and identity element , and consider its
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
consisting of all
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities 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 subsetsIn mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group (mathematics), group ''G'', then their product is the subset of ''G'' defined by :ST = \. The subsets ''S'' and ''T'' need not be su ...
. * Let be a set. The set of all functions forms a monoid under
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. 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 (mathematics), function that always returns the same value that was ... . It is also called the ''
full transformation monoidIn algebra, a transformation semigroup (or composition semigroup) is a collection of Transformation (function), transformations (function (mathematics), functions from a set to itself) that is closure (mathematics), closed under function composition. ...
'' 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 Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
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 group ...
s of , denoted , forms a monoid under composition of
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ... s. For more on the relationship between category theory and monoids see below. * The set of
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
classes Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of
compact surface In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only co ...
s with the
connected sum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ... . 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) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ... , 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 :: $f\left(i\right) := \begin i+1, & \text 0 \le i < n-1 \\ k, & \text i = n-1. \end$ :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 The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... 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 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 A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
.

## 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
(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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, then its Grothendieck group is the
trivial group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.

## 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. 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 system In theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as wel ...
s of semiautomata. A
transformation semigroup In algebra, a transformation semigroup (or composition semigroup) is a collection of Transformation (function), transformations (function (mathematics), functions from a set to itself) that is closure (mathematics), closed under function composition ...
can be made into an operator monoid by adjoining the identity transformation.

# Monoid homomorphisms A
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
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 In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', ...
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 #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... , as it necessarily preserves the identity (because, in a group, the identity is the only element such that ). A
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
monoid homomorphism is called a monoid
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... . 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 (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...
. One does this by specifying a set of generators Σ, and a set of relations on the
free monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
Σ. One does this by extending (finite)
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
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 In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic m ...
, and : $\langle a,b \,\vert\; aba=baa, bba=bab\rangle$ is the
plactic monoidIn mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableau, Young tableaux. It was discovered by (who called it the t ...
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 Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
. Indeed, the axioms required of a monoid operation are exactly those required of
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ... 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... s between single object categories. So this construction gives an equivalence of categories, equivalence between the category of monoids, category of (small) monoids Mon and a full subcategory of the category of (small) categories Cat. Similarly, the category of groups 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 (category theory), monoid object which is an abstract definition of what is a monoid in a category. A monoid object in category of sets, Set is just a monoid.

# Monoids in computer science

In computer science, many abstract data types can be endowed with a monoid structure. In a common pattern, a sequence of elements of a monoid is "fold (higher-order function), 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 parallelization, 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 ''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 can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a binary tree might differ depending on pre-order vs. post-order tree traversal.

# MapReduce

An application of monoids in computer science is 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 operation 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 Finitary, infinitary sum operation $\Sigma_I$ for any index set such that:Droste, 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 : $\sum_ = \sum_\left(m_i\right)\quad \text \bigcup_ I_j=I \text I_j \cap I_ = \emptyset \quad \text j\neq j\text{'}$ A continuous monoid is an ordered commutative monoid in which every directed set has a least upper bound compatible with the monoid operation: : $a + \sup S = \sup\left(a + S\right) \ .$ These two concepts are closely related: a continuous monoid is a complete monoid in which the infinitary sum may be defined as : $\sum_I a_i = \sup \sum_E a_i$ where the supremum on the right runs over all finite subsets of and each sum on the right is a finite sum in the monoid.