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. 
For example, the functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. 
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes theory), and 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 , which we will denote •, is a monoid if it satisfies the following two axioms:
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. 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 and . 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 is a subset "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 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 is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the 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 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 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 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 trace monoid; trace monoids commonly occur in the theory of concurrent computation.
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 formula_13 of any sequence formula_14 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.
Grothendieck group construction.
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 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 , which isn't true. A monoid has the cancellation property (or is cancellative) if for all "a", "b" and "c" in "M", always implies and always implies . A commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck 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. Proof: Fix an element "x" in the monoid. Since the monoid is finite, for some . But then, by cancellation we have that where "e" is the identity. Therefore, , 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.
It turns out that requiring the cancellative property in a monoid is not required to perform the Grothendieck construction – commutativity is sufficient. However, if the original monoid has an absorbing element then its Grothendieck group is the trivial group. Hence the homomorphism is, in general, not injective.
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:
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 and is a function such that
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. For example, consider formula_15, the set of residue classes modulo formula_16 equipped with multiplication. In particular, the class of formula_17 is the identity. Function formula_18 given by formula_19 is a semigroup homomorphism as formula_20 in formula_21. However, formula_22, 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 ).
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 , 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 formula_23. Thus, for example,
is the equational presentation for the bicyclic monoid, and
is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as formula_26 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,
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 functors between single object categories. 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.
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 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 formula_27 and associative operation formula_28, the "fold" operation is defined as follows:
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 (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 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 infinitary sum operation formula_30 for any index set such that:
and
A continuous monoid is an ordered commutative monoid in which every directed set has a least upper bound compatible with the monoid operation:
These two concepts are closely related: a continuous monoid is a complete monoid in which the infinitary sum may be defined as
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.