Encyclopedia - Isomorphism: Encyclopedia Ii - Isomorphism - Definition
Douglas Hofstadter provides an informal definition: The word "isomorphism" applies when two complex structures can be mapped onto each o...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Informal Discussion
Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting ...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Homomorphism For Beginners
Homomorphism is one of the fundamental concepts in abstract algebra. Because abstract algebra studies sets with operations that generate ...   » Read the article

Encyclopedia - Connectedness: Encyclopedia Ii - Connectedness - Other Notions Of Connectedness
There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish t...   » Read the article

Encyclopedia - Category Theory: Encyclopedia - Category Theory
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is...   » Read the article

Encyclopedia - Coherent Sheaf: Encyclopedia - Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a...   » Read the article

Encyclopedia - Monoid: Encyclopedia - Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an ident...   » Read the article

Encyclopedia - Zero Morphism: Encyclopedia - Zero Morphism
In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in ...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia - Sheaf Mathematics
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each ope...   » Read the article

Encyclopedia - Analogy: Encyclopedia - Analogy
Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particu...   » Read the article

Encyclopedia - Alexander Grothendieck: Encyclopedia - Alexander Grothendieck
Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of ...   » Read the article

Encyclopedia - Connectedness: Encyclopedia - Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object ...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia - Category Mathematics
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories...   » Read the article

Encyclopedia - Abelian Variety: Encyclopedia - Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be...   » Read the article

Encyclopedia - Isomorphism: Encyclopedia Ii - Isomorphism - Two Abstract Examples
Isomorphism - A relation-preserving isomorphism. For example, if one object consists of a set X with an ordering ≤ and the other obje...   » Read the article

Encyclopedia - Isomorphism: Encyclopedia Ii - Isomorphism - Applications
Group isomorphism is where the objects in question are groups. Similarly, if the objects are fields, it is called a field isomorphism. In...   » Read the article

Encyclopedia - Isomorphism: Encyclopedia Ii - Isomorphism - Practical Example
The following is an example of an isomorphism from ordinary algebra. Consider the logarithm function: For any fixed base b, the logarithm...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Kernel Of A Homomorphism
Main article: kernel (algebra) Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Homomorphism For Mathematicians
In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant str...   » Read the article

Encyclopedia - Vector Bundle: Encyclopedia - Vector Bundle
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Formal Definition
A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properti...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Types Of Homomorphisms
The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the...   » Read the article

Encyclopedia - Continuous Function Topology: Encyclopedia - Continuous Function Topology
In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which pre...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Types Of Homomorphisms
The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the...   » Read the article

Encyclopedia - Vector Bundle: Encyclopedia Ii - Vector Bundle - Definition And First Consequences
A real vector bundle is given by the following data: topological spaces X (the "base space") and E (the "total space") a continuous map ...   » Read the article

Encyclopedia - Image Category Theory: Encyclopedia Ii - Image Category Theory - Examples
In the category of sets the image of a morphism is the inclusion from the ordinary image to Y. In many concrete categories such as grou...   » Read the article

Encyclopedia - Groupoid: Encyclopedia Ii - Groupoid - Definitions
From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, ...   » Read the article

Encyclopedia - Group Action: Encyclopedia Ii - Group Action - Orbits And Stabilizers
Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of ...   » Read the article

Encyclopedia - Gluing Axiom: Encyclopedia Ii - Gluing Axiom - Removing Restrictions On C
To rephrase this definition in a way that will work in any category C that has sufficient structure, we note that we can write the object...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Definition
Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case,...   » Read the article

Encyclopedia - Integral Domain: Encyclopedia Ii - Integral Domain - Examples
The prototypical example is the ring Z of all integers. Every field is an integral domain. Conversely, every Artinian integral domain is ...   » Read the article

Encyclopedia - Initial Object: Encyclopedia Ii - Initial Object - Properties
Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - The Polynomial Ring In Several Variables
Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered...   » Read the article

Encyclopedia - Lattice Order: Encyclopedia Ii - Lattice Order - Formal Definition
As mentioned above, lattices can be characterized both as posets and as algebraic structures. Both approaches and their relationship are ...   » Read the article

Encyclopedia - Ringed Space: Encyclopedia Ii - Ringed Space - Morphisms
A morphism of ringed spaces is simply a morphism of sheaves. Explicitly, a morphism from (X, OX) to (Y, OY) is given by the following dat...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Definition
An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (whe...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Relation To Lie Groups
Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a ...   » Read the article

Encyclopedia - Automorphism: Encyclopedia Ii - Automorphism - Definition
The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "iso...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Mathematical Definition Of A Function
A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an eleme...   » Read the article

Encyclopedia - Natural Transformation: Encyclopedia Ii - Natural Transformation - Examples
Natural transformation - A worked example. Statements like "Every group is naturally isomorphic to its opposite group" abound in mode...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Introduction
The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler...   » Read the article

Encyclopedia - Cartesian Closed Category: Encyclopedia Ii - Cartesian Closed Category - Examples
Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The pro...   » Read the article

Encyclopedia - Metric Space: Encyclopedia Ii - Metric Space - Definition
A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function d : X × X → R such that d(x, y)...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - The Vocabulary Of Functions
An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the ...   » Read the article

Encyclopedia - Monoid: Encyclopedia Ii - Monoid - Properties
Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an e...   » Read the article

Encyclopedia - Abelian Variety: Encyclopedia Ii - Abelian Variety - Analytic Theory
Abelian variety - Definition. A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex ma...   » Read the article

Encyclopedia - Continuous Function Topology: Encyclopedia Ii - Continuous Function Topology - Definitions
Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous functi...   » Read the article

Encyclopedia - Category Theory: Encyclopedia Ii - Category Theory - Background
The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider t...   » Read the article

Encyclopedia - Logical Graph: Encyclopedia Ii - Logical Graph - Formal Development
What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopeful...   » Read the article

Encyclopedia - Normal Morphism: Encyclopedia Ii - Normal Morphism - Definition
A category C must have zero morphisms for the concept of normality to make complete sense. In that case, we say that a monomorphism is no...   » Read the article

Encyclopedia - Monoid: Encyclopedia Ii - Monoid - Properties
Directly from the definition, one can show that the identity element e is unique. Then it is possible to define invertible elements: an e...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia Ii - Category Mathematics - Definition
A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target ...   » Read the article

Encyclopedia - Analogy: Encyclopedia Ii - Analogy - Models And Theories Of Analogy
Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathemat...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Facts And Logic
The concept of "average profit" suggested that a process of competition and market-balancing had already established a uniform (or ruling...   » Read the article

Encyclopedia - Alexander Grothendieck: Encyclopedia Ii - Alexander Grothendieck - Mathematical Achievements
Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been inven...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Introduction
Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary w...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Properties
Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X ...   » Read the article

Encyclopedia - Proper Map: Encyclopedia Ii - Proper Map - Algebraic Varieties And Schemes
Proper map - Definition. A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber pr...   » Read the article

Encyclopedia - Category Theory: Encyclopedia Ii - Category Theory - Universal Constructions, Limits, And Colimits
Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be...   » Read the article

Encyclopedia - Category Theory: Encyclopedia Ii - Category Theory - Categories, Objects, And Morphisms
Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a ...   » Read the article

Encyclopedia - Connectedness: Encyclopedia Ii - Connectedness - Connectivity
Properties and parameters based on the idea of connectedness often involve the word connectivity. For example, in graph theory, a connect...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Is A Function More Than Its Graph?
Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codo...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Partial Functions And Multi-functions
The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x i...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Image Of A Set
One often extends the concept (and notation) of image of an argument to sets of arguments. Namely, if A is any subset of the domain X, th...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Set Of All Functions
The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Composing Functions
The functions f: X â†’ Y and g: Y â†’ Z can be composed by first applying f to an argument x and then apply...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Specifying A Function
If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions With Multiple Inputs And Outputs
Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combinatio...   » Read the article

Encyclopedia - Automorphism: Encyclopedia Ii - Automorphism - Automorphism Group
The set of automorphisms of an object X form a group under composition of morphisms. This group is called the automorphism group of X. Th...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Three Types Of Production Prices
Most Marxists missed the fact that Marx identified (though often not very clearly) three main types of production prices: the private or...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Two Interpretations Of Production Prices
Unfortunately, Marx never prepared the manuscript of the third volume of Das Kapital for publication. Therefore his draft text, which ske...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Production Prices And The Transformation Problem
The concept of production prices is one "building block" in Marx's theory of the "equalising tendency of the rate of profit", which aimed...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Value And Price
A lot of criticism of Marx's concept originates from the ambiguities referred to earlier. Consequently, many of the criticisms can be dis...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - History Of The Concept
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 ...   » Read the article

Encyclopedia - Proper Map: Encyclopedia Ii - Proper Map - Topological Spaces
Proper map - Definition. A function f : X → Y between two topological spaces is proper if and only if the preimage of every comp...   » Read the article

Encyclopedia - Prices Of Production: Encyclopedia Ii - Prices Of Production - Basic Definition
For most political economists, this kind of price corresponds roughly to Adam Smith's concept of "natural prices" and the modern neoclass...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Formal Definition
Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consis...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Other Fields
Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Domain Codomain Argument Image
If f is a function from X to Y, the set X is called the domain of f, and Y is called its codomain. Each element of the domain is called a...   » Read the article

Encyclopedia - Automorphism: Encyclopedia Ii - Automorphism - Inner And Outer Automorphisms
In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two classes. In the case of ...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Examples
We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles ...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Classes Of Functions
Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-o...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Graph Of A Function
The graph of a function f is the set of all ordered pairs (x, f(x)), for all x in the domain X. If X and Y are the set of real numbers (o...   » Read the article

Encyclopedia - Cartesian Closed Category: Encyclopedia Ii - Cartesian Closed Category - Applications
In cartesian closed categories, a "function of two variables" (a morphism f:X×Y → Z) can always be represented as a "function of one v...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Coalgebras
An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (pr...   » Read the article

Encyclopedia - Logical Graph: Encyclopedia Ii - Logical Graph - Computational Representation
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 me...   » Read the article

Encyclopedia - Integral Domain: Encyclopedia Ii - Integral Domain - Divisibility Prime And Irreducible Elements
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 t...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - The Polynomial Ring R[x]
One can then check that the set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication ...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - Definition Of A Polynomial
In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynom...   » Read the article

Encyclopedia - Groupoid: Encyclopedia Ii - Groupoid - Relation To Groups
If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is litera...   » Read the article

Encyclopedia - Logical Graph: Encyclopedia Ii - Logical Graph - Quick Tour Of The Neighborhood
This much preparation allows us to present the two most basic axioms of logical graphs, shown in graph and string forms below, along with...   » Read the article

Encyclopedia - Integral Domain: Encyclopedia Ii - Integral Domain - Characteristic And Homomorphisms
The characteristic of every integral domain is either zero or a prime number. If R is an integral domain with prime characteristic p, the...   » Read the article

Encyclopedia - Logical Graph: Encyclopedia Ii - Logical Graph - Duality Logical And Topological
There are two types of duality that have to be kept separately mind in the use of logical graphs, logical duality and topological duality...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Homomorphisms Subalgebras And Ideals
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), Ï†(y)]...   » Read the article

Encyclopedia - Logical Graph: Encyclopedia Ii - Logical Graph - In Lieu Of A Beginning
In medias res, as always, we nevertheless need a quantum of formal matter to keep the topical momentum going. A game try at supplying tha...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Classification Of Lie Algebras
Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Examples
1. Every vector space becomes an abelian Lie algebra trivially if we define the Lie bracket to be identically zero. 2. Euclidean space R3...   » Read the article

Encyclopedia - Lie Algebra: Encyclopedia Ii - Lie Algebra - Definition
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 [·, Â...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Examples
Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fi...   » Read the article

Encyclopedia - Kernel Category Theory: Encyclopedia Ii - Kernel Category Theory - Relation To Other Categorical Concepts
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 v...   » Read the article

Encyclopedia - Groupoid: Encyclopedia Ii - Groupoid - Examples
From linear algebra: Given a field K, the general linear groupoid GL*(K) consists of all invertible matrices with entries from K, with co...   » Read the article