In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. Monoids occur in a number of branches of mathematics and capture the idea of function composition; indeed, this notion is abstracted in category theory, where the monoid is a category with one object. Monoids are also commonly used to provide an 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. Some of the more important results in the study of monoids are the Krohn-Rhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups.

Definition

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

• Associativity: for all a, b, c in M, (a*b)*c = a*(b*c)
• Identity element: there exists an element e in M, such that for all a in M, a*e = e*a = a.

One often sees the additional axiom

• Closure: for all a, b in M, a*b is in M

though, strictly speaking, this axiom is implied by the notion of a binary operation.

Alternatively, a monoid is a semigroup with an identity element; it is also be understood as a category with closure.

A monoid satisfies all the axioms of a group with the exception of having inverses. 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. 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. 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). 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. 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. 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.

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

• Every singleton set {x} gives rise to a one-element (trivial) monoid. For fixed x this monoid is unique, since the monoid axioms require that x*x = x in this case.
• Every group is a monoid and every abelian group a commutative monoid.
• Every bounded semilattice is an idempotent commutative monoid.
• Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e*e = e and e*s = s = s*e for all s ∈ S.
• The natural numbers, N, form a commutative monoid under addition (identity element zero), or multiplication (identity element one). A submonoid of N under addition is called a numerical monoid.
• The elements of any unital ring, with addition or multiplication as the operation.
• The integers, rational numbers, real numbers or complex numbers, with addition or multiplication as operation.
• The set of all n by n matrices over a given ring, with matrix addition or matrix multiplication as the operation.
• The set of all finite strings over some fixed alphabet Σ forms a monoid with string concatenation as the operation. The empty string serves as the identity element. This monoid is denoted Σ^∗ and is called the free monoid over Σ.
• Fix a monoid M, and consider its power set P(M) consisting of all subsets of M. A binary operation for such subsets can be defined by S * T = {s * t : s in S and t in T}. This turns P(M) into a monoid with identity element {e}. In the same way the power set of a group G is a monoid under the product of group subsets.
• Let S be a set. The set of all functions S → S forms a monoid under function composition. The identity is just the identity function. If S is finite with n elements, the monoid of functions on S is finite with nⁿ elements.
• Generalizing the previous example, let C be a category and X an object in C. The set of all endomorphisms of X, denoted End_C(X), forms a monoid under composition of morphisms. For more on the relationship between category theory and monoids see below.
• The set of homeomorphism classes of compact surfaces with the connected sum. Its unit element is the class of the ordinary 2-sphere. Furthermore, if a denotes the class of the torus, and b denotes the class of the projective plane, then every element c of the monoid has a unique expression the form c=na+mb where n is the integer ≥ 0 and m=0,1, or 2. We have 3b=a+b.
• Let $$ be a cyclic monoid of order n, that is, $=\; \{f^0,f^1,..,f^\{n-1\}\}$. Then $f^n\; =\; f^k$ for some $0\; le\; k\; le\; n$. In fact, each such k gives a distinct monoid of order n, and every cyclic monoid is isomorphic to one of these.

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 & kend{bmatrix}

or, equivalently

Multiplication of elements in $$ 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.

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 x^n+p=xⁿ * x^p 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. 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 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. A commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck construction. 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). 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. Proof: Fix an element x in the monoid. Since the monoid is finite, xⁿ = x^m for some m > n > 0. But then, by cancellation we have that x^m-n = e where e is the identity. Therefore x * x^m-n-1 = e, so x has an inverse.

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.

Acts and operator monoids

Let M be a monoid. Then a (left) M-act (or left act over M) is a set X together with an operation • : M × X → X which is compatible with the monoid structure as follows:

• for all x in X: e • x = x;
• for all a, b in M and x in X: a • (b • x) = (a * b) • x.

This is the analogue in monoid theory of a (left) group action. Right M-acts are defined in a similar way. A monoid with an act is also known as an operator monoid. Important examples include transition systems of semiautomata. A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.

Monoid homomorphisms

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

• f(x*y) = f(x)•f(y) for all x, y in M
• f(e) = e′

where e and e′ are the identities on M and M′ respectively. Monoid homomorphisms are sometimes simply called monoid morphisms.

Not every semigroup homomorphism is a monoid homomorphism since it may not preserve the identity. Contrast this with the case of group homomorphisms: the axioms of group theory ensure that every semigroup homomorphism between groups preserves the identity. 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. 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. That is, it is a subset

$sim;subseteq\; Mtimes\; M$

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

A monoid congruence induces congruence classes

$[m]\; =\; \{xin\; Mvert;\; xsim\; m\}$

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

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

which is a monoid homomorphism. 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

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

Several additional notations are common. Give a subset $Lsubseteq\; M$, one writes

$[L]\; =\; \{[m]\; ,vert;\; min\; L\}$

for the set of congruence classes induced by L. In this notation, clearly $[M]=M/sim$. In general, however, $[L]$ is not a monoid. Going in the opposite direction, if $Xsubseteq\; [M]$ 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 general, $bigcup\; X$ is not a monoid.

Clearly, one has $Lsubseteq\; bigcup[L]$ and $left[bigcup\; Xright]=X$.

Equational presentation

Monoids may be given a presentation, much in the same way that groups can be specified by means of a group presentation. One does this by specifying a set of generators Σ, and a set of relations on the free monoid Σ^∗. One does this by extending (finite) binary relations on Σ^∗ to monoid congruences, and then constructing the quotient monoid, as above.

Given a binary relation R ⊂ Σ^∗ × Σ^∗, one defines its symmetric closure as R ∪ R⁻¹. This can be extended to a symmetric relation E ⊂ Σ^∗ × Σ^∗ by defining x ~_E y if and only if x = sut and y = svt for some strings u, v, s, t ∈ Σ^∗ with (u,v) ∈ R ∪ 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,

$langle\; p,q,vert;\; pq=1rangle$

is the equational presentation for the bicyclic monoid, and

$langle\; a,b\; ,vert;\; aba=baa,\; bba=babrangle$

is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as $a^ib^j(ba)^k$ 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. 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. 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 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.

See also

• Star height problem
• Kleene algebra

References

• John M. Howie, Fundamentals of Semigroup Theory (1995), Clarendon Press, Oxford ISBN 0-19-851194-9
• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3110152487.