morphism

The parse graphs that we've been looking at so far are one step toward the pointer graphs that it takes to make trees live in computer memory, but they are still a couple of steps too abstract to properly suggest in much concrete detail the species of dynamic data structures that we need. The time has come to flesh out the skeleton that we've drawn up to this point. Nodes in a graph depict records in computer memory. A record is a collection of data that can be thought to reside at a specific address. For semioticians, a ...

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Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

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If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b. If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference. The elements which divide 1 are called the units of R; these are ...

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Integral domain, Integral domain - Examples, Integral domain - Divisibility prime and irreducible elements, Integral domain - Field of fractions, Integral domain - Characteristic and homomorphisms

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One can then check that the set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication mentioned above, forms itself a ring, the polynomial ring over R, which is denoted by R[X]. Formally these two ring operations are functions defined on with values in R[X], given by the formulas and If ...

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Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings

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In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynomial function. This definition cannot be adapted to a general ring, however. For example, over the ring Z/2Z of integers modulo 2, the polynomial P(X)=X2+X=X(X+1) takes only the value 0, as when k is an integer, k(k+1) is always even. But we would expec ...

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Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings

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If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory can be generalized to groupoids, with the notion of group homomorphism being replaced by that of functor. If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic, with an isomorphism given ...

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Groupoid, Groupoid - Definitions, Groupoid - Examples, Groupoid - Relation to groups, Groupoid - Covariance in special relativity, Groupoid - Lie groupoids and Lie algebroids

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This much preparation allows us to present the two most basic axioms of logical graphs, shown in graph and string forms below, along with handy names for referring to the two different directions of applying the axioms. o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` | | ` ` ` ...

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Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

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The characteristic of every integral domain is either zero or a prime number. If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism f : R -> R, the Frobenius homomorphism. ...

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Integral domain, Integral domain - Examples, Integral domain - Divisibility prime and irreducible elements, Integral domain - Field of fractions, Integral domain - Characteristic and homomorphisms

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There are two types of duality that have to be kept separately mind in the use of logical graphs, logical duality and topological duality. There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its topological genus — can be represented in linear text as what are called parse strings or traversal strings< ...

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Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

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A homomorphism φ : g → h between Lie algebras g and h over the same base field F is an F-linear map such that [φ(x), φ(y)] = φ([x, y]) for all x and y in g. The composition of such homomorphisms is again a homomorphism, and the Lie algebras over the field F, together with these morphisms, form a category. If such a homomorphism is bijective, it is called an isomorphism, and the two Lie algebras g and h are called isomorphic ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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In medias res, as always, we nevertheless need a quantum of formal matter to keep the topical momentum going. A game try at supplying that least bit of motivation may be found in this duo of transformations between the indicated forms of enclosure: o-----------------------------------------------------------o | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | o------- ...

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Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

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Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of Lie groups. Every finite-dimensional real or complex Lie algebra arises as the Lie algebra of unique real or complex simply connected Lie group (Ado's theorem), but there may be more than one group, even more than one connected group, giving rise to the same algebra. For instance, the groups SO(3) (3×3 orthogonal matrices of determinant 1) and SU(2) (2×2 unitary matrices of determinant 1) both give rise to the sam ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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1. Every vector space becomes an abelian Lie algebra trivially if we define the Lie bracket to be identically zero. 2. Euclidean space R3 becomes a Lie algebra with the Lie bracket given by the cross product of vectors. 3. If an associative algebra A with multiplication * is given, it can be turned into a Lie algebra by defining [x, y] = x * y − y * x. This expression is called the commutator of x and y. ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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A Lie algebra is a type of an algebra over a field; it is a vector space g over some field F together with a binary operation [·, ·] : g × g → g, called the Lie bracket, which satisfies the following properties: Bilinearity: for all a, b ∈ F and all x, y, z ∈ g. For all x in g < ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field). To be explicit, if f : X → Y is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of X and the inclusion homomorphism from KSee also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

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The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In s ...

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Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

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From linear algebra: Given a field K, the general linear groupoid GL*(K) consists of all invertible matrices with entries from K, with composition given by matrix multiplication. If G = GL*(K), then G0 contains a copy of the set of natural numbers, since there is one identity matrix of dimension n for each natural number n, although G0 contains other matrices. G(m,n) is empty unless m = n, in which case it is the ...

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Groupoid, Groupoid - Definitions, Groupoid - Examples, Groupoid - Relation to groups, Groupoid - Covariance in special relativity, Groupoid - Lie groupoids and Lie algebroids

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The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. Ordinary functions are sometimes referred to as morphisms in a concrete category. ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the late 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy N ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope at a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - History of the concept

Using the language of category theory, a Lie algebra can be defined as an object A in the category of vector spaces together with a morphism such that where and σ is the cyclic permutation braiding . In diagrammatic form: ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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The category C is called cartesian closed iff it satisfies the following three properties: it has a terminal object any two objects X and Y of C have a product X×Y in C any two objects Y and Z of C have an exponential ZY in C For the first two conditions above, it is the same to require that any finite (possibly empty) family of objects of C admit a product in C, because of t ...

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Cartesian closed category, Cartesian closed category - Definition, Cartesian closed category - Examples, Cartesian closed category - Applications, Cartesian closed category - Equational theory

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