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Cylindric Algebra
In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the Algebraic logic, algebraization of first-order logic with equality. This is comparable to the role Boolean algebra (structure), Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model Quantification (logic), quantification and equality (mathematics), equality. They differ from polyadic algebras in that the latter do not model equality. The cylindric algebra should not be confused with the measure theory, measure theoretic concept ''cylindrical algebra'' that arises in the study of cylinder set measures and the cylindrical σ-algebra. Definition of a cylindric algebra A cylindric algebra of dimension \alpha (where \alpha is any ordinal number) is an algebraic structure (A,+,\cdot,-,0,1,c_\kappa,d_)_ such that (A,+,\cdot,-,0,1) is a Boolean algebra (structure), Boolean algebra, c_\kappa a ... [...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] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Henkin
Leon Albert Henkin (April 19, 1921, Brooklyn, New York – November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the Type theory, theory of types. He was an active scholar at the University of California, Berkeley, University of California, Berkeley, where he made great contributions as a researcher and teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the ''Group in Logic and the Methodology of Science'', from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defender of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notre Dame Journal Of Formal Logic
The ''Notre Dame Journal of Formal Logic'' is a quarterly peer-reviewed scientific journal covering the foundations of mathematics and related fields of mathematical logic, as well as philosophy of mathematics. It was established in 1960 and is published by Duke University Press on behalf of the University of Notre Dame. The editors-in-chief are Curtis Franks and Anand Pillay (University of Notre Dame). The founder of the magazine was Boleslaw Sobocinski. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2012 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 0.431. References External links * Journal pageat Notre Dame University Journal pageat Project Euclid Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Algebraic Decomposition
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, along with an algorithm to compute it, that is fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decomposition is a decomposition of R''n'' into connected semialgebraic sets called ''cells'', on which each polynomial has constant sign, either +, − or 0. To be ''cylindrical'', this decomposition must satisfy the following condition: If 1 ≤ ''k'' < ''n'' and ''π'' is the projection from R''n'' onto R''n''−''k'' consisting in removing the last ''k'' coordinates, then for every pair of cells ''c'' and ''d'', one has either ''π''(''c'') = ''π''(''d'') or ''π''(''c'') ∩ ''π''(''d'') = ∅. This implies that the images by ''π'' of the cells define a cylindrical decomposition of R''n''−''k''. The notion was introduced by |
Polyadic Algebra
Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tarski algebra). There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras (when equality is part of the logic) and Lawvere's functorial semantics (a categorical approach). References Further reading *Paul Halmos, ''Algebraic Logic'', Chelsea Publishing The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on on ..., New York (1962) Algebraic logic {{mathlogic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation Algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X'' 2 of all binary relations on a set ''X'', that is, subsets of the cartesian square ''X''2, with ''R''•''S'' interpreted as the usual composition of binary relations ''R'' and ''S'', and with the converse of ''R'' as the converse relation. Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminated in the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed by Alfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself be conducted without variables. Definition A relation algebra is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperdoctrine
In mathematics, specifically category theory, a doctrine is roughly a system of theories ("categorical analogues of fragments of logical theories which have sufficient category-theoretic structure for their models to be described as functors"). For example, an algebraic theory, as invented by William Lawvere, is an example of a doctrine. The concept of doctrines was invented by Lawvere as part of his work on algebraic theories. The name is based on a suggestion by Jon Beck.Marquis, Jean-Pierre, and Gonzalo Reyes"The history of categorical logic: 1963-1977."(2004). A doctrine can be defined in several ways: * as a 2-monad. This was Lawvere's original approach. * as a 2-category; the idea is that each object there amounts to a "theory". * As cartesian double theories, as logics, or as a class of limits. References Further reading * ''Generalised algebraic models'', by Claudia Centazzo. * William Lawvere, Ordinal sums and equational doctrines, Lecture Notes in Math., Vol. 80 (Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatory Logic
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments. In mathematics Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebraic Logic
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 2003. History The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebra (structure), Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and then further developed and refined by others, especially Charles Sanders Peirce, C. S. Peirce and Ernst Schröder (mathematician), Ernst Schröder, from the 1870s to the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
