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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that Monoids occur in a number of branches of mathematics. In geometry, a monoid captures the idea of function composition; indeed, this notion is abstracted in category theory, where the monoid is a category with one object. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Monoids are also commonly used to lay a firm algebraic foundation for computer science; in this case, the transition monoid and syntactic monoid are used in describing a finite state machine, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output In Mathematics, the syntactic monoid M ( L) of a Formal language L is the smallest Monoid that recognizes the In Mathematics and Computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute but others are not In Mathematics and Computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection In Computer science, the process calculi (or process algebras) are a diverse family of related approaches to formally modelling Concurrent systems Process Concurrent computing is the concurrent (simultaneous execution of multiple interacting computational tasks Some of the more important results in the study of monoids are the Krohn-Rhodes theorem and the 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 The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation

Contents

Definition

A monoid is a set M with binary operation * : M × M → M, obeying the following axioms:

One often sees the additional axiom

though, strictly speaking, this axiom is implied by the notion of a binary operation. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

Alternatively, a monoid is a semigroup with an identity element. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

A monoid satisfies all the axioms of a group with the exception of having inverses. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to A monoid with inverses is a group.

By abuse of notation we sometimes refer to M itself as a monoid, implying the presence of identity and operation. In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition A monoid can be denoted by the tuple (M, *) if the operation needs to be made explicit.

Generators and submonoids

A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x,y∈N then x*y∈N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N=N*, where the superscript * is the Kleene star. In Mathematical logic and Computer science, the Kleene star (or Kleene closure) is a Unary operation, either on sets of For any subset N of M, the monoid N* is the smallest monoid that contains N.

A subset N is said to be a generator of M if and only if M=N*. If N is finite, then M is said to be finitely generated.

Commutative monoid

A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). In Mathematics, commutativity is the ability to change the order of something without changing the end result Commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions An order-unit of a commutative monoid M is an element u of M such that for any element x of M, there exists a positive integer n such that x ≤ nu. This is often used in case M is the positive cone of a partially ordered abelian group G, in which case we say that u is an order-unit of G. In Abstract algebra, an ordered group is a group (G+ equipped with a Partial order "≤" which is translation-invariant In Abstract algebra, an ordered group is a group (G+ equipped with a Partial order "≤" which is translation-invariant There is an algebraic construction that will take any commutative monoid, and turn it into a full-fledged abelian group; this construction is known as the Grothendieck group. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the Grothendieck group construction in Abstract algebra constructs an Abelian group from a Commutative Monoid in the

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. In Mathematics and Computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute but others are not Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often

Acts, transition systems

An operator monoid is a monoid M which acts upon a set X. In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output That is, there is an operation • : M × X → X which is compatible with the monoid operation.

Operator monoids are also known as acts (since they resemble a group action), transition systems, semiautomata or transformation semigroups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Theoretical computer science, a state transition system is an Abstract machine used in the study of Computation. In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output

Examples

Moreover, f can be considered as a function on the points 0,1,2,. . ,n − 1 given by

\begin{bmatrix} 0 & 1 & 2 & ... & n-2 & n-1 \\ 1 & 2 & 3 & ... & n-1 & k\end{bmatrix}

or, equivalently

f(i) := \begin{cases} i+1, & \mbox{if } 0 \le i < n-1 \\ k, & \mbox{if } i = n-1. \end{cases}

Multiplication of elements in < f > is then given by function composition.

Note also that when k = 0 then the function f is a permutation of {0,1,2,. . ,n − 1} and gives the unique cyclic group of order n. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

Properties

In a monoid, one can define positive integer powers of an element x : x1=x, and xn=x*. . . *x (n times) for n>1 . The rule of powers xn+p=xn * xp is obvious.

Directly from the definition, one can show that the identity element e is unique. Then, for any x , one can set x0=e and the rule of powers is still true with nonnegative exponents.

It is possible to define invertible elements: an element x is called invertible if there exists an element y such x*y = e and y*x = e. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to The element y is called the inverse of x . Associativity guarantees that inverses, if they exist, are unique.

If y is the inverse of x , one can define negative powers of x by setting x−1=y and x−n=y*. . . *y (n times) for n>1 . And the rule of exponents is still verified for all n,p rational integers. This is why the inverse of x is usually written x−1. The set of all invertible elements in a monoid M, together with the operation *, forms a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In that sense, every monoid contains a group (if only the trivial one consisting of the identity alone).

However, not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements a and b exist such that a*b = a holds even though b is not the identity element. Such a monoid cannot be embedded in a group, because in the group we could multiply both sides with the inverse of a and would get that b = e, which isn't true. A monoid (M,*) has the cancellation property (or is cancellative) if for all a, b and c in M, a*b = a*c always implies b = c and b*a = c*a always implies b = c. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. A commutative monoid with the cancellation property can always be embedded in a group. That's 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). In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation 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. obviously include the identity and not so obviously are closed under the operation). This means that the cancellative elements of any commutative monoid can be extended to a group.

An inverse monoid, is a monoid where for every a in M, there exists a unique a-1 in M such that a=a*a-1*a and a-1=a-1*a*a-1. If an inverse monoid is cancellative, then it is a group.

Monoid homomorphisms

A homomorphism between two monoids (M,*) and (M′,•) is a function f : M → M′ such that

where e and e′ are the identities on M and M′ respectively. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector Monoid homomorphisms are sometimes simply called monoid morphisms.

Not every magma (groupoid) homomorphism is a monoid homomorphism since it may not preserve the identity. In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. Contrast this with the case of group homomorphisms: the axioms of group theory ensure that every magma (groupoid) homomorphism between groups preserves the identity. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. For monoids this isn't always true and it is necessary to state it as a separate requirement.

A bijective monoid homomorphism is called a monoid isomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Two monoids are said to be isomorphic if there is an isomorphism between them.

Monoid congruence and the quotient monoid

A monoid congruence is an equivalence relation that is compatible with the monoid product. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" That is, it is a subset

\sim\;\subseteq M\times M

such that it is reflexive, symmetric and transitive (just as every equivalence relation must be), and also has the property that if x\sim y\, and u\sim v\, for every x,y,u and v in M, then one has that x*u\sim y*v\,.

A monoid congruence induces congruence classes

[m] = \{x\in M\vert\; x\sim m\}

and the monoid operation * induces a binary operation \circ on the congruence classes:

[u]\circ [v] = [u*v]

which is a monoid homomorphism. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers It is also clearly associative, and so the set of all congruence classes are a monoid as well. This monoid is called the quotient monoid, and may be written as

M/\sim\; = \{[m]\,\vert\; m\in M\}.

Several additional notations are common. Give a subset L\subseteq M, one writes

[L] = \{[m] \,\vert\; m\in L\}

for the set of congruence classes induced by L. In this notation, clearly [M] = M / ˜. In general, however, [L] is not a monoid. Going in the opposite direction, if X\subseteq [M] is a subset of the quotient monoid, one writes

\bigcup X = \{m \,\vert\; [m]\in X\}.

This is, of course, just the set-theoretic union of the members of X. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In general, \bigcup X is not a monoid.

Clearly, one has L\subseteq \bigcup[L] and \left[\bigcup X\right]=X.

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, one method of defining a group is by a presentation. One does this by specifying a set of generators Σ, and a set of relations on the free monoid Σ * . In Abstract algebra, the free monoid on a set A is the Monoid whose elements are all the finite sequences (or strings) of zero or One does this by extending (finite) binary relations on Σ * to monoid congruences, and then constructing the quotient monoid, as above. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of

Given a binary relation R\subseteq \Sigma^*\times\Sigma^*, one defines its symmetric closure as R\cup R^{-1}. This can be extended to a symmetric relation E\subseteq \Sigma^*\times\Sigma^* by defining x\sim_E y\, if and only if x = sut and y = svt for some strings s,t\in \Sigma^* and (u,v)\in R\cup R^{-1}. 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=\{u_1=v_1,\cdots,u_n=v_n\}. 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). 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 Elements of this plactic monoid may be written as aibj(ba)k for integers i,j,k, as the relations show that ba commutes with both a and b.

Relation to category theory

Group-like structures
Totality Associativity Identity Division
Group Yes Yes Yes Yes
Monoid Yes Yes Yes No
Semigroup Yes Yes No No
Loop Yes No Yes Yes
Quasigroup Yes No No Yes
Magma Yes No No No
Groupoid No Yes Yes Yes
Category No Yes Yes No

Monoids can be viewed as a special class of categories. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Mathematics, especially in Category theory and Homotopy theory In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and That is,

A monoid is, essentially, the same thing as a category with a single object.

More precisely, given a monoid (M,*), 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 functors between single object categories. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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.

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. In Category theory, a monoid (or monoid object) (M\mu\eta in a Monoidal category C is an object M together with two

See also

References

Dictionary

monoid

-noun

  1. (mathematics) A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.
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