Monoid

A set S equipped with a binary operation S × S → 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.[1] 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.

• Out of the 16 possible binary Boolean operators, four have a two-sided identity that is also commutative and associative. These four each make the set {False, True} a commutative monoid. Under the standard definitions, AND and XNOR have the identity True while XOR and OR have the identity False. The monoids from AND and OR are also idempotent while those from XOR and XNOR are not.
• The set of natural numbers N = {0, 1, 2, ...} is a commutative monoid under addition (identity element 0) or multiplication (identity element 1). A submonoid of N under addition is called a numerical monoid.
• The set of positive integers 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 semilattice is an 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 semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e • s = s = s • e for all s ∈ S. This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids.[5]
  • Thus, an idempotent monoid (sometimes known as find-first) may be formed by adjoining an identity element e to the left zero semigroup over a set S. The opposite monoid (sometimes called find-last) is formed from 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.[6]
  • 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. 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, M₁, ..., M_k), their Cartesian product M × N is also a monoid (respectively, M₁ × ⋅⋅⋅ × M_k). The associative operation and the identity element are defined pairwise.[7]
• 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 ∈ S, t ∈ 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. It is also called the full transformation monoid of S. 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 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 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 a positive integer and m = 0, 1, or 2. We have 3b = a + b.
• Let ⟨f⟩ be a cyclic monoid of order n, that is, ⟨f⟩ = {f⁰, f¹, ..., fⁿ⁻¹}. Then fⁿ = f^k for some 0 ≤ 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

$${\displaystyle {\begin{bmatrix}0&1&2&\cdots &n-2&n-1\\1&2&3&\cdots &n-1&k\end{bmatrix}}}$$

or, equivalently

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

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.

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.

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 ⋅ : 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.

Example monoid homomorphism x ↦ 2^x from (N, +, 0) to (N, ×, 1). It is injective, but not surjective.

A homomorphism between two monoids (M, ∗) and (N, •) is a function f : M → 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 semigroup homomorphism 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.[11] For example, consider M_n, the set of residue classes modulo n equipped with multiplication. In particular, the class of 1 is the identity. Function f : M₃ → M₆ given by f(k) = 3k is a semigroup homomorphism as 3k ⋅ 3l = 9kl = 3kl in M₆. However, f(1) = 3 ≠ 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, as it necessarily preserves the identity (because, in a group, the identity is the only element such that x ⋅ 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.

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 = {u₁ = v₁, ..., u_n = v_n}. Thus, for example,

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

is the equational presentation for the bicyclic monoid, and

${\displaystyle \langle a,b\,\vert \;aba=baa,bba=bab\rangle }$

is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as ${\displaystyle a^{i}b^{j}(ba)^{k}}$ for integers i, j, k, as the relations show that ba commutes with both a and b.

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.[12] 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.[12] So this construction gives an equivalence between the category of (small) monoids Mon and a full subcategory of the category of (small) categories Cat. Similarly, the category of groups 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.[12]

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.

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 "folded" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be parallelized by employing a prefix sum or similar algorithm, in order to utilize multiple cores or processors efficiently.

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

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

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.

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.

A complete monoid is a commutative monoid equipped with an infinitary sum operation ${\displaystyle \Sigma _{I}}$ for any index set I such that[13][14][15][16]

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

and

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

An ordered commutative monoid is a commutative monoid M together with a partial ordering ≤ such that a ≥ 0 for every a ∈ M, and a ≤ b implies a + c ≤ b + c for all a, b, c ∈ M.

A continuous monoid is an ordered commutative monoid (M, ≤) in which every directed subset has a least upper bound, and these least upper bounds are compatible with the monoid operation:

${\displaystyle a+\sup S=\sup(a+S)}$

for every a ∈ M and directed subset S of M.

If (M, ≤) is a continuous monoid, then for any index set I and collection of elements (a_i)_i∈I, one can define

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

and M together with this infinitary sum operation is a complete monoid.[16]

• Cartesian monoid
• Green's relations
• Monad (functional programming)
• Semiring and Kleene algebra
• Star height problem
• Vedic square
• Frobenioid

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• Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander V. (2000), Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers, de Gruyter Expositions in Mathematics, vol. 29, Berlin: Walter de Gruyter, ISBN 3-11-015248-7, Zbl 0945.20036
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