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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 algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and then further developed and refined by others, especially C. S. Peirce and 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) went on to discover cylindric algebra, whose representable instances algebraize a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unary Operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation: :3 − −2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the resu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''. History Before the nineteenth century, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Donald Pigozzi
Donald is a masculine given name derived from the Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the Gaelic pronunciation by English speakers, and partly associated with the spelling of similar-sounding Germanic names, such as ''Ronald''. A short form of ''Donald'' is '' Don''. Pet forms of ''Donald'' include ''Donnie'' and ''Donny''. The feminine given name ''Donella'' is derived from ''Donald''. ''Donald'' has cognates in other Celtic languages: Modern Irish ''Dónal'' (anglicised as ''Donal'' and ''Donall'');. Scottish Gaelic ''Dòmhnall'', ''Domhnull'' and ''Dòmhnull''; Welsh '' Dyfnwal'' and Cumbric ''Dumnagual''. Although the feminine given name ''Donna'' is sometimes used as a feminine form of ''Donald'', the names are not etymologically related. Variations Kings and noblemen Domnall or Domhnall is the name of many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wim Blok
Willem Johannes "Wim" Blok (1947–2003) was a Dutch logician who made major contributions to algebraic logic, universal algebra, and modal logic. His important achievements over the course of his career include "a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic." Blok began his career in 1973 as an algebraist investigating the varieties of interior algebras at the University of Illinois at Chicago. Following the 1976 completion of his Ph.D. on that topic, he continued on to study more general varieties of modal algebras. As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s. He published many papers in the '' Reports on Mathematical Logic'', served as a member on their editorial board, and was one of their guest editors. Along with Don Pigozzi, Wim Blok co-authored the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Janusz Czelakowski
Janusz () is a masculine Polish given name. It is also the shortened form of January and Januarius. People * Janusz Akermann (born 1957), Polish painter * Janusz Bardach, Polish gulag survivor and physician *Janusz Bielański, Roman Catholic priest *Janusz Bojarski (born 1956), Polish general * Janusz Bokszczanin (1894–1973), Polish Army colonel *Janusz Christa (1934–2008), Polish author of comic books *Janusz Domaniewski (1891–1954), Polish ornithologist *Janusz Gajos, Polish actor * Janusz Gaudyn (1935–1984), Polish physician, writer and poet *Janusz Głowacki (1938–2017), Polish-American author and screenwriter * Janusz Janowski (born 1965), Polish painter, jazz drummer and art theorist *Janusz Kamiński (born 1959), Polish cinematographer and film director *Janusz Korczak (Henryk Goldszmit), Polish-Jewish children's author, pediatrician, and child pedagogist *Janusz Kurtyka (born 1960), Polish historian specializing in the culture and religion of Poland in the 16th a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Roman Suszko
Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *''Epistle to the Romans'', shortened to ''Romans'', a letter in the New Testament of the Christian Bible Roman or Romans may also refer to: Arts and entertainment Music * Romans (band), a Japanese pop group * ''Roman'' (album), by Sound Horizon, 2006 * ''Roman'' (EP), by Teen Top, 2011 *"Roman (My Dear Boy)", a 2004 single by Morning Musume Film and television *Film Roman, an American animation studio * ''Roman'' (film), a 2006 American suspense-horror film * ''Romans'' (2013 film), an Indian Malayalam comedy film * ''Romans'' (2017 film), a British drama film * ''The Romans'' (''Doctor Who''), a serial in British TV series People *Roman (given name), a given name, including a list of people and fictional characters *Roman (surname), including a list of people named Roman or Romans *Ῥωμα� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Jerzy Łoś
Jerzy Łoś (born 22 March 1920 in Lwów, Poland (now Lviv, Ukraine) – 1 June 1998 in Warsaw) () was a Polish mathematician, logician, economist, and philosopher. He is especially known for his work in model theory, in particular for " Łoś's theorem", which states that any first-order formula is true in an ultraproduct if and only if it is true in "most" factors (see ultraproduct for details). In model theory he also proved many preservation theorems, but he gave significant contributions, as well, to foundations of mathematics, Abelian group theory and universal algebra. In the 60's he turned his attention to mathematical economics, focusing mainly on production processes and dynamic decision processes. He was faculty at academies in Wrocław, Toruń, and Warsaw Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |