Monoid

In

associative binary operation and an identity element. For example, the nonnegative integers

with addition form a monoid, the identity element being

0

.

Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.

The functions from a set into itself form a monoid with respect to function composition. More generally, in

object

to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.

In

process calculi and concurrent computing

.

In

formal language theory (star height problem

).

See semigroup for the history of the subject, and some other general properties of monoids.

Definition

A set $S$ equipped with a binary operation $S \times S \to S$ , 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 $(a • b) • c = a • (b • c)$ holds.
Identity element: There exists an element $e$ in $S$ such that for every element $a$ in $S$ , the equalities $e • a = a$ and $a • e = a$ hold.

In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.^[a] For this reason the identity is regarded as a constant, i. e. $0$ -ary (or nullary) operation. The monoid therefore is characterized by specification of the triple $(S, • , e)$ .

Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written $(ab) c = a (bc)$ and $ea = ae = a$ . 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 $(M, •)$ is a subset $N$ of $M$ that is closed under the monoid operation and contains the identity element $e$ of $M$ .^[1]^[b] Symbolically, $N$ is a submonoid of $M$ if $e \in N \subseteq M$ , and $x • y \in N$ whenever $x, y \in N$ . 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

nonnegative integers

.

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

positive cone of a partially ordered abelian group

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

concurrent computation

.

Examples

Out of the 16 possible

idempotent

while those from XOR and XNOR are not.

N ∖ {0}

is a commutative monoid under multiplication (identity element

1

).

Given a set

A

, the set of subsets of

A

is a commutative monoid under intersection (identity element is

A

itself).

Given a set $A$ , the set of subsets of $A$ is a commutative monoid under union (identity element is the empty set).
Generalizing the previous example, every bounded
idempotent
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 algebras are endowed with these monoid structures.
Every
singleton set ${x}$ 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 a commutative monoid.
Any
free functor between the category of semigroups and the category of monoids.^[3]
- Thus, an idempotent monoid (sometimes known as find-first) may be formed by adjoining an identity element e to the
  right zero semigroup
  over S.
  - Adjoin an identity $e$ to the left-zero semigroup with two elements ${lt, gt}$ . Then the resulting idempotent monoid ${lt, e, gt}$ models the
    lexicographical order
    of a sequence given the orders of its elements, with e representing equality.

The underlying set of any
ring
, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity $1$ .)
The integers, rational numbers, real numbers or complex numbers, with addition or multiplication as operation.^[4]

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
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 $Σ$ . It is not commutative if $Σ$ has at least two elements.

Given any monoid

M

, the opposite monoid

M op

has the same carrier set and identity element as

M

, and its operation is defined by

x • op y = y • x

. Any commutative monoid is the opposite monoid of itself.

Given two sets $M$ and $N$ endowed with monoid structure (or, in general, any finite number of monoids, $M 1, ..., M k$ ), their Cartesian product $M \times N$ , with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, $M 1 \times \cdot\cdot\cdot \times M k$ ).^[5]
Fix a monoid $M$ . The set of all functions from a given set to $M$ is also a monoid. The identity element is a constant function mapping any value to the identity of $M$ ; the associative operation is defined pointwise.
Fix a monoid $M$ with the operation $•$ and identity element $e$ , and consider its power set $P (M)$ consisting of all subsets of $M$ . A binary operation for such subsets can be defined by $S • T = {s • t : s \in S, t \in T}$ . 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 $S \to S$ forms a monoid under
full transformation monoid
of $S$ . If $S$ is finite with $n$ elements, the monoid of functions on $S$ is finite with $n n$ elements.
Generalizing the previous example, let $C$ be a category and $X$ an object of $C$ . 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
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 a positive integer and $m = 0, 1$ , or $2$ . We have $3 b = a + b$ .

Let

⟨ f ⟩

be a cyclic monoid of order

n

, that is,

⟨ f ⟩ = {f 0, f 1, ..., f n -1}

. Then

f n = f k

for some

0 \leq k < 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&\cdots &n-2&n-1\\1&2&3&\cdots &n-1&k\end{bmatrix}}

or, equivalently

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

Multiplication of elements in $⟨ f ⟩$ is then given by function composition.

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

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

Products and powers

For each nonnegative integer $n$ , one can define the product $p_{n}=\textstyle \prod _{i=1}^{n}a_{i}$ of any sequence $(a 1, ..., a n)$ of $n$ elements of a monoid recursively: let $p 0 = e$ and let $p m = p m -1 • a m$ for $1 \leq m \leq n$ .

As a special case, one can define nonnegative integer powers of an element $x$ of a monoid: $x 0 = 1$ and $x n = x n -1 • x$ for $n \geq 1$ . Then $x m + n = x m • x n$ for all $m, n \geq 0$ .

Invertible elements

An element $x$ is called invertible if there exists an element $y$ such that $x • y = e$ and $y • x = e$ . The element $y$ is called the inverse of $x$ . Inverses, if they exist, are unique: if $y$ and $z$ are inverses of $x$ , then by associativity $y = ey = (zx) y = z (xy) = ze = z$ .^[6]

If $x$ is invertible, say with inverse $y$ , then one can define negative powers of $x$ by setting $x - n = y n$ for each $n \geq 1$ ; this makes the equation $x m + n = x m • x n$ hold for all $m, n \in Z$ .

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 $a$ 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 multiplying both sides with the inverse of $a$ would get that $b = e$ , which is not true.

A monoid $(M, •)$ has the cancellation property (or is cancellative) if for all $a$ , $b$ and $c$ in $M$ , the equality $a • b = a • c$ implies $b = c$ , and the equality $b • a = c • a$ implies $b = c$ .

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.^[c]

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 $a • b = a • c$ , then $b$ and $c$ have the same image in the Grothendieck group, even if $b \neq c$ . 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 $a$ 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.

In the opposite direction, a zerosumfree monoid is an additively written monoid in which $a + b = 0$ implies that $a = 0$ and $b = 0$ :^[7] 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 $\cdot : M \times X \to X$ which is compatible with the monoid structure as follows:

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

This is the analogue in monoid theory of a (left)

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 $(N, •)$ is a function $f : M \to N$ such that

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

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

Not every

residue classes

modulo

n

equipped with multiplication. In particular,

[1] n

is the identity element. Function

f : [Z] 3 \to [Z] 6

given by

[k] 3 \mapsto [3 k] 6

is a semigroup homomorphism, since

[3 k \cdot 3 l] 6 = [9 kl] 6 = [3 kl] 6

. However,

f ([1] 3) = [3] 6 \neq [1] 6

, 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, as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only element $x$ such that $x \cdot x = x$ ).

A

bijective monoid homomorphism is called a monoid isomorphism

. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.

Equational presentation

Monoids may be given a presentation, much in the same way that groups can be specified by means of a

group presentation. One does this by specifying a set of generators

Σ

, and a set of relations on the free monoid

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 $R \subset Σ * \times Σ *$ , one defines its symmetric closure as $R \cup R -1$ . This can be extended to a symmetric relation $E \subset Σ * \times Σ *$ by defining $x ~ E y$ if and only if $x = sut$ and $y = svt$ for some strings $u, v, s, t \in Σ *$ with $(u, v) \in R \cup R -1$ . 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 $R = {u 1 = v 1, ..., u n = v n}$ . 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). Elements of this plactic monoid may be written as $a^{i}b^{j}(ba)^{k}$ 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^α	Associativity	Identity	Divisibility^β	Commutativity
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Associative quasigroup	Required	Required	Unneeded	Required	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required
^α The closure axiom, used by many sources and defined differently, is equivalent. ^β Here, divisibility refers specifically to the quasigroup axioms.

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.^[8] 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

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.[8]

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

parallelized by employing a prefix sum

Given a sequence of values of type $M$ with identity element $ε$ and associative operation $•$ , the fold operation is defined as follows:

\mathrm {fold} :M^{*}\rightarrow M=\ell \mapsto {\begin{cases}\varepsilon &{\mbox{if }}\ell =\mathrm {nil} \\m\bullet \mathrm {fold} \,\ell '&{\mbox{if }}\ell =\mathrm {cons} \,m\,\ell '\end{cases}}

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 the so-called MapReduce programming model (see 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 $I$ such that^[9]^[10]^[11]^[12]

\sum _{i\in \emptyset }{m_{i}}=0;\quad \sum _{i\in \{j\}}{m_{i}}=m_{j};\quad \sum _{i\in \{j,k\}}{m_{i}}=m_{j}+m_{k}\quad {\text{ for }}j\neq k

and

\sum _{j\in J}{\sum _{i\in I_{j}}{m_{i}}}=\sum _{i\in I}m_{i}\quad {\text{ if }}\bigcup _{j\in J}I_{j}=I{\text{ and }}I_{j}\cap I_{j'}=\emptyset \quad {\text{ for }}j\neq j'

.

An ordered commutative monoid is a commutative monoid $M$ together with a

partial ordering

\leq

such that

a \geq 0

for every

a \in M

, and

a \leq b

implies

a + c \leq b + c

for all

a, b, c \in M

.

A continuous monoid is an ordered commutative monoid $(M, \leq)$ in which every

least upper bound

, and these least upper bounds are compatible with the monoid operation:

a+\sup S=\sup(a+S)

for every $a \in M$ and directed subset $S$ of $M$ .

If $(M, \leq)$ is a continuous monoid, then for any index set $I$ and collection of elements $(a i) i \in I$ , one can define

\sum _{I}a_{i}=\sup _{{\text{finite }}E\subset I}\;\sum _{E}a_{i},

and $M$ together with this infinitary sum operation is a complete monoid.^[12]

Notes

^ If both $e 1$ and $e 2$ satisfy the above equations, then $e 1 = e 1 • e 2 = e 2$ .
^ Some authors omit the requirement that a submonoid must contain the identity element from its definition, requiring only that it have an identity element, which can be distinct from that of $M$ .
^ Proof: Fix an element $x$ in the monoid. Since the monoid is finite, $x n = x m$ 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.
^ $f (x) * f (e M) = f (x * e M) = f (x)$ for each $x$ in $M$ , when $f$ is a semigroup homomorphism and $e M$ is the identity of its domain monoid $M$ .