|
Esakia Duality
In mathematics, Esakia duality is the equivalence of categories, dual equivalence between the category (mathematics), category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces. Let Esa denote the category of Esakia spaces and Esakia space#Esakia morphisms, Esakia morphisms. Let be a Heyting algebra, denote the set of prime ideal, prime filters of , and denote set-theoretic inclusion on the prime filters of . Also, for each , let , and let denote the topology on generated by . Theorem: is an Esakia space, called the ''Esakia dual'' of . Moreover, is a Heyting algebra isomorphism from onto the Heyting algebra of all clopen set, clopen up-sets of . Furthermore, each Esakia space is isomorphic in Esa to the Esakia dual of some Heyting algebra. This representation of Heyting algebras by means of Esakia spaces is functorial and yields a dual equivalence between the categories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Esakia Space
In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974. Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality—the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Definition For a partially ordered set and for , let and let . Also, for , let and . An Esakia space is a Priestley space such that for each clopen subset of the topological space , the set is also clopen. Equivalent definitions There are several equivalent ways to define Esakia spaces. Theorem: Given that is a Stone space, the following conditions are equivalent: :(i) is an Esakia space. :(ii) is closed for each and is clopen for each clopen . :(iii) is closed for each and for each (where denotes the closure in ). :(iv) is closed for each , the least closed set containing an up-set is an up-set, and the least up-set containing a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. Prime ideals for commutative rings An ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ideal in \Z. Examples * A simple example: In the ring R=\Z, the subset of even numbers is a prime ideal. * Given an integral domain R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clopen Set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as " door spaces" their name. Examples In any topological space X, the empty set and the whole space X are both clopen. Now consider the space X which consists of the union of the two o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up-set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger than ''s'' (that is, if s \leq x), then ''x'' is in ''S''. In words, this means that any ''x'' element of ''X'' that is \,\geq\, to some element of ''S'' is necessarily also an element of ''S''. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset ''S'' of ''X'' with the property that any element ''x'' of ''X'' that is \,\leq\, to some element of ''S'' is necessarily also an element of ''S''. Definition Let (X, \leq) be a preordered set. An in X (also called an , an , or an set) is a subset U \subseteq X that is "closed under going up", in the sense that :for all u \in U and all x \in X, if u \leq x then x \in U. The dual notion is a ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functorial
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphisms
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Spaces
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological space and let ''K''\circ(''X'') be the set of all compact open subsets of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions: *''X'' is compact and T0. * ''K''\circ(''X'') is a basis of open subsets of ''X''. * ''K''\circ(''X'') is closed under finite intersections. * ''X'' is sober, i.e., every nonempty irreducible closed subset of ''X'' has a (necessarily unique) generic point. Equivalent descriptions Let ''X'' be a topological space. Each of the following properties are equivalent to the property of ''X'' being spectral: #''X'' is homeomorphic to a projective limit of finite T0-spaces. #''X'' is homeomorphic to the spectrum of a bounded distributive lattice ''L''. In this case, ''L'' is is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
