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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Partially ordered set
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In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for pairs of elements in the set. The word partial in the partial order or partially ordered set is used as an indication that not every pair of elements need be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset, Partial orders thus generalize total orders, in which every pair is comparable. To be an order, a binary relation must be reflexive, antisymmetric. One familiar example of an ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, a poset can be visualized through its Hasse diagram, which depicts the ordering relation. A partial order is a binary relation ≤ over a set P satisfying particular axioms which are discussed below, when a ≤ b, we say that a is related to b. The axioms for a partial order state that the relation ≤ is reflexive, antisymmetric. That is, for all a, b, and c in P, it must satisfy, in other words, a partial order is an antisymmetric preorder. A set with an order is called a partially ordered set. The term ordered set is also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as ordered sets, for a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. In the figure on top-right, e. g. and are comparable, while and are not, a partial order under which every pair of elements is comparable is called a total order or linear order, a totally ordered set is also called a chain. A subset of a poset in which no two elements are comparable is called an antichain. A more concise definition will be given using the strict order corresponding to ≤. For example, is covered by in the figure. Standard examples of posets arising in mathematics include, The real numbers ordered by the standard less-than-or-equal relation ≤, the set of subsets of a given set ordered by inclusion

3.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed

4.
Monoid
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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as 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 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 science, both in its foundational aspects and in practical programming. The set of strings built from a set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history provide a foundation for process calculi. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem, the history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. Identity element There exists an element e in S such that for every element a in S, 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 monoid in which each element has an inverse is a group. Depending on the context, the symbol for the operation may be omitted, so that the operation is denoted by juxtaposition, for example. This notation does not imply that it is numbers being multiplied, N is thus a monoid under the binary operation inherited from M. If there is a generator of M that has finite cardinality, 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, commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y 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 cone of a partially ordered abelian group 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

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Morphism
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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics, in set theory, morphisms are functions, in linear algebra, linear transformations, in group theory, group homomorphisms, in topology, continuous functions, and so on. The study of morphisms and of the structures over which they are defined is central to category theory, in category theory, morphisms are sometimes also called arrows. A category C consists of two classes, one of objects and the other of morphisms, there are two objects that are associated to every morphism, the source and the target. For many common categories, objects are sets and morphisms are functions from an object to another object, therefore, the source and the target of a morphism are often called domain and codomain respectively. A morphism f with source X and target Y is written f, X → Y, thus a morphism is represented by an arrow from its source to its target. Morphisms are equipped with a binary operation, called composition. The composition of two morphisms f and g is defined if and only if the target of f is the source of g, the source of g∘f is the source of f, and the target of g∘f is the target of g. Associativity h ∘ = ∘ f whenever the operations are defined, that is when the target of f is the source of g, for a concrete category, the identity morphism is just the identity function, and composition is just the ordinary composition of functions. Associativity then follows, because the composition of functions is associative, the composition of morphisms is often represented by a commutative diagram. For example, The collection of all morphisms from X to Y is denoted homC or simply hom, some authors write MorC, Mor or C. Note that the term hom-set is something of a misnomer as the collection of morphisms is not required to be a set, a category where hom is a set for all objects X and Y is called locally small. Note that the domain and codomain are in part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, the two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes hom be disjoint, in practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms. A morphism f, X → Y is called a monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 and it is also called a mono or a monic. A morphism f has an inverse if there is a morphism g, Y → X such that g ∘ f = idX. The left inverse g is called a retraction of f

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Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are said to represent a process connecting two objects, or in many cases a structure-preserving transformation connecting two objects. There are, however, many applications where more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences