In abstract algebraa branch of mathematicsa monoid is an algebraic structure with a single associative binary operation and an identity element.

Monoids are studied in semigroup theory, because they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set.

In fact, all functions from a set into itself form naturally a monoid with respect to function composition. Monoids are also commonly used in computer scienceboth in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid.

The transition monoid and syntactic monoid are used in describing finite state machineswhereas 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. 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. This notation does not imply that it is numbers being multiplied. N is thus a monoid under the binary operation inherited from M. A subset S of M is said to be a generator of M if M is the smallest set containing S that is closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S.

If there is a generator of M that has finite cardinality, then M is said to be finitely generated. Not every set S will generate a monoid, as the generated structure may lack an identity element. A monoid whose operation is commutative is called a commutative monoid or, less commonly, an abelian monoid.

Commutative monoids are often written additively. This is often used in case M is the positive cone of a partially ordered abelian group Gin which case we say that u is an order-unit of G. 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 a monoid, one can define positive integer powers of an element x: From the definition of a monoid, one can show that the identity element e is unique.

It is possible to define invertible elements: The element y is called the inverse of x. Thus inverses, if they exist, are unique. And the rule of exponents is still verified for all np rational integers. In that sense, every monoid contains a group possibly only the trivial group consisting of only the identity. However, not every monoid sits inside a group.

A commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck construction. However, a non-commutative cancellative monoid need not be embeddable in a group. If a monoid has the cancellation property and is finitethen it is in fact a group. Fix an element x in the monoid. The right- and left-cancellative elements of a monoid each in turn form a submonoid i. 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 option straddle trades, not injective. If an inverse monoid is cancellative, then it is a group. 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 are sometimes simply called monoid morphisms. Not every semigroup homomorphism is a monoid homomorphism, since it may not how to invest money in walmart stock the identity to the identity of the target monoid, even though the element it maps the identity to will be an identity of the image of the mapping.

In contrast, a semigroup homomorphism between groups is always a group homomorphismas it necessarily preserves the identity. Number commutative binary operations set for monoids this isn't always true, it is necessary to amazon binary option bots work this as a separate requirement.

A bijective leading indicators for forex trading 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 presentationmuch in the same way that groups can be specified by means of a group presentation. Finally, one takes the reflexive and transitive closure of Ewhich is then a monoid congruence. 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. Likewise, monoid homomorphisms are just functors between single object categories. Similarly, click ads and earn money in india without investment 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, Monwhose 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.

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.

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. Probably the most important application of monoids in computer science is so-called MapReduce programming model see Encoding Map-Reduce As A Monoid With Left Folding.

MapReducein computing, consists of two or three operations. Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. As described above, this operation can be highly parallelized. The only problem is that the monoid may be pretty large, so it is distributed through the cloud, and the results of fold cannot be stored in one node. If this is the case, an operation of "Shuffling" is added, to group results of mapping. For example, if we have a multisetin a program it is represented as a map from elements to their numbers.

Elements are called keys in this case.

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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. 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: From Wikipedia, the free encyclopedia. For other uses, see Monoid disambiguation. Not to be confused with monad.

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Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra. Presentation of a monoid. Graphs, Dioids and Semirings: New Models and Algorithms. A New ApproachSpringer Monographs in Mathematics, 71Springer, p.

Pacific Journal of Mathematics. Semirings and Formal Power Series. Handbook of Weighted Automata3— Bayreuther Mathematische Schriften in German. In Paterson, Michael S. Automata, Languages and Programming: Lecture Notes in Computer Science. Algebraic foundations in computer science.

Essays dedicated to Symeon Bozapalidis on the occasion of his retirement.

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