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Duality Theory For Distributive Lattices
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'' o ... [...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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...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 profileat 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 morphi ... [...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 (closed) basis consisting of all sets of the form \, where ''b'' is an element of ''B''. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For every Boolean algebra ''B'', ''S''(''B'') is a compact totall ... [...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. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called ''fields of sets'', closely related to the ''rings of sets'' used by Birk ... [...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 th ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
