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Priestley's Representation Theorem For Distributive Lattice
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let . Then is a spectral space, where the topology on is generated by . The spectral space is called the ''prime spectrum'' of . The map is a lattice isomorphism from onto the lattice of all compact open subsets of . In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice. Similarly, if and denotes the topology generated by , then is also a spectral space. Moreover, is a pairwise Stone space. The pairwise Stone space is called the '' bitopological dual'' of . Eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitopological Space
In mathematics, a bitopological space is a set endowed with ''two'' topologies. Typically, if the set is X and the topologies are \sigma and \tau then the bitopological space is referred to as (X,\sigma,\tau). The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric. Continuity A map \scriptstyle f:X\to X' from a bitopological space \scriptstyle (X,\tau_1,\tau_2) to another bitopological space \scriptstyle (X',\tau_1',\tau_2') is called continuous or sometimes pairwise continuous if \scriptstyle f is continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ... both as a map from \scriptstyle (X,\tau_1) to \scriptstyle (X',\tau_1') and as map from \scriptstyle (X,\tau_2) to \scriptstyle (X',\tau_2'). Bito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proc
Proc may refer to: * Proč, a village in eastern Slovakia * '' Proč?'', a 1987 Czech film * procfs or proc filesystem, a special file system (typically mounted to ) in Unix-like operating systems for accessing process information * Protein C (PROC) * Proc, a term in video game terminology * Procedures or process, in the programming language ALGOL 68 * People's Republic of China, the formal name of China * the official acronym for the Canadian House of Commons Standing Committee on Procedure and House Affairs * People's Republic of the Congo The People's Republic of the Congo () was a Marxist–Leninist socialist state that existed in the Republic of the Congo from 1969 to 1992. The People's Republic of the Congo was founded in December 1969 as the first Marxist-Leninist state ... * Pro*C, a programming language {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bull
A bull is an intact (i.e., not Castration, castrated) adult male of the species ''Bos taurus'' (cattle). More muscular and aggressive than the females of the same species (i.e. cows proper), bulls have long been an important symbol cattle in religion and mythology, in many religions, including for sacrifices. These animals play a significant role in beef ranching, dairy farming, and a variety of sporting and cultural activities, including bullfighting and bull riding. Due to their temperament, handling of bulls requires precautions. Nomenclature The female counterpart to a bull is a cow, while a male of the species that has been Castration, castrated is a ''steer'', ''Oxen, ox'', or ''bullock'', although in North America, this last term refers to a young bull. Use of these terms varies considerably with area and dialect. Colloquially, people unfamiliar with cattle may also refer to steers and heifers as "cows", and bovines of aggressive or long-horned breeds as "bulls" reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilary Priestley
Hilary Ann Priestley is a British mathematician. She is a professor at the University of Oxford and a Fellow of St Anne's College, Oxford, where she has been Tutor in Mathematics since 1972. Hilary Priestley introduced ordered separable topological spaces; such topological spaces are now usually called Priestley spaces in her honour. The term " Priestley duality" is also used for her application of these spaces in the representation theory of distributive lattices. Books * *Reviews of ''Introduction to Lattices and Order'': T. S. Blyth, , ; Jonathan Cohen, ''ACM SIGACT News'', ; Amy Davidow, ''Amer. Math. Monthly'', ; Josef Niederle, ; Václav Slavík, * References External links Hilary Priestley home pageProfessor Hilary Priestley profile at the Mathematical Institute, University of Oxford Professor Hilary Ann Priestley profileat St Anne's College, Oxford Hilary Priestleyon ResearchGate ResearchGate is a European commercial social networking site for scientists and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Esakia Duality
In mathematics, Esakia duality is the dual equivalence between the 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 morphisms. Let be a Heyting algebra, denote the set of 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 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 * HA of Heyting algebras and Heyting algebra homomorphisms and * Esa of Esakia spaces and Esakia morp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Representation Theorem For Boolean Algebras
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a basis consisting of all sets of the form \, where ''b'' is an element of ''B''. These sets are also closed and so are clopen (both closed and open). This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Representation Theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (other).'' In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. Here, a lattice is an abstract structure with two binary operations, the "meet" and "join" operations, which must obey certain axioms; it is distributive if these two operations obey the distributive law. The union and intersection operations, in a family of sets that is closed under these operations, automatically form a distributive lattice, and Birkhoff's representation theorem states that (up to isomorphism) every finite distributive lattice can be formed in this way. It is named after Garrett Birkhoff, who published a proof of it in 1937.. The theorem can be interpreted as providing a one-to-one correspondenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure. Examples Algebra * Cayley's theorem states that every group is isomorphic to a permutation group. * Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces. *Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. *: A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set. *: Another variant, Stone's duality, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the categories of Boolean algebras and that of Stone spaces. * The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DL Duality
DL, dL, or dl may stand for: Science and technology Electronics and computing *, HTML element#dl, an HTML element used for a definition list *Deep learning, a field of machine learning *Description logics, a family of knowledge representation languages *Delete Line (ANSI), an ANSI X3.64 escape sequence *Digital library, a library in which collections are stored in digital formats *Diode logic, a logic family using diodes *DVD-R DL, a DVD Dual Layer engineering method *DL register, the low byte of an X86 16-bit DX register *Dynamic loading, a mechanism for a computer program to load a library Telecommunications *Data link, a computer connection for transmitting data *Distribution list, a function of e-mail clients *Downlink, the link from a satellite to a ground station *Download, a transfer of electronic data Vehicles *Subaru DL, an automobile *Australian National DL class, a class of diesel locomotives built by Clyde Engineering *New Zealand DL class locomotive, a diesel-el ... [...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 Category (mathematics), 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 dual (category theory), 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 betwe ... [...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] |
