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Green's Relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility. Instead of working directly with a semigroup ''S'', it is convenient to define Green's relations over the monoid ''S''1. (''S''1 is "''S'' with an identity adjoined if necessary"; if ''S'' is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by so ... [...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] |
Unit (ring Theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More generally, any root of unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Ideal
Relative may refer to: General use * Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives'' Philosophy *Relativism, the concept that points of view have no absolute truth or validity, having only relative, subjective value according to differences in perception and consideration, or relatively, as in the relative value of an object to a person * Relative value (philosophy) Economics *Relative value (economics) Popular culture Film and television * ''Relatively Speaking'' (1965 play), 1965 British play * ''Relatively Speaking'' (game show), late 1980s television game show * ''Everything's Relative'' (episode)#Yu-Gi-Oh! (Yu-Gi-Oh! Duel Monsters), 2000 Japanese anime ''Yu-Gi-Oh! Duel Monsters'' episode *'' Relative Values'', 2000 film based on the play of the same name. *''It's All Relative'', 2003-4 comedy television series *''Intelligence is Relative'', tag line for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs are ri''n''gs without ''n''egative elements, similar to using '' rng'' to mean a r''i''ng without a multiplicative ''i''dentity. Tropical semirings are an active area of research, linking algebraic varieties with piecewise linear structures. Definition A semiring is a set R equipped with two binary operations \,+\, and \,\cdot,\, called addition and multiplication, such that:Lothaire (2005) p.211Sakarovitch (2009) pp.27–28 * (R, +) is a commutative monoid with identity element 0: ** (a + b) + c = a + (b + c) ** 0 + a = a = a + 0 ** a + b = b + a * (R, \,\cdot\,) is a monoid with identity element 1: ** (a \cdot b) \cdot c = a \cdot (b \cdot c) ** 1 \cdot a = a = a \cdot 1 * Multiplication left and right distributes over addition: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Small Groups
The following list in mathematics contains the finite groups of small order up to group isomorphism. Counts For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is : 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... For labeled groups, see . Glossary Each group is named by their Small Groups library as G''o''''i'', where ''o'' is the order of the group, and ''i'' is the index of the group within that order. Common group names: * Z''n'': the cyclic group of order ''n'' (the notation C''n'' is also used; it is isomorphic to the additive group of Z/''n''Z). * Dih''n'': the dihedral group of order 2''n'' (often the notation D''n'' or D2''n'' is used ) ** K4: the Klein four-group of order 4, same as and Dih2. * S''n'': the symmetric group of degree ''n'', containing the ''n''! permutations of ''n'' elements. * A''n'': the alternating group of degree ''n'', containing the even permutations of ''n'' elements, of order 1 for , and order ''n''!/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalent (semigroup Theory)
Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivalents'', a series of photographs of clouds by Alfred Stieglitz Language *Dynamic and formal equivalence in translation *Equivalence (formal languages) Law *The doctrine of equivalents in patent law *The equivalence principle as if impacts on the direct effect of European Union law Logic *Logical equivalence, where two statements are logically equivalent if they have the same logical content *Material equivalence, a relationship where the truth of either one of the connected statements requires the truth of the other Science and technology Chemistry *Equivalent (chemistry) *Equivalence point *Equivalent weight Computing *Turing equivalence (theory of computation), or Turing completeness *Semantic equivalence in computer metadata Economic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicyclic Semigroup
In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic monoid describing the Dyck language of balanced pairs of parentheses. Thus, it finds common applications in combinatorics, such as describing binary trees and associative algebras. History The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H. Clifford and Gordon Preston claim that one of them, working with David Rees, discovered it independently (without publication) at some point before 1943. Construction There are at least three standard ways of constructing the bicyclic semigroup, and various notations for referring to it. Lyapin called it ''P''; Clifford and Preston used \mathcal; and most recent papers have tended to use ''B''. This article will use the modern style throughout. From a free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon Preston
Gordon Bamford Preston (28 April 1925 – 14 April 2015) was an English mathematician best known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from Magdalen College, Oxford. He was born in Workington and brought up in Carlisle. During World War II, he left his undergraduate studies at Oxford University for Bletchley Park, to help crack German codes with a small group of mathematicians, which included Alan Turing. At Bletchley Park he persuaded Max Newman (who thought that the women would not care for the "intellectual effort") to authorise talks to the Wrens to explain their work mathematically, and the talks were very popular. After graduation, he was a teacher at Westminster School, London and then The Royal Military College of Science. In 1954 he wrote three highly influential papers in the Journal of the London Mathematical Society, laying the foundations of inverse semigroup theory. Before Preston and Alfred H. Clifford's book, ''The Algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epigroup
In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all ''x'' in a semigroup ''S'', there exists a positive integer ''n'' and a subgroup ''G'' of ''S'' such that ''x''''n'' belongs to ''G''. Epigroups are known by wide variety of other names, including quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr), or just π-regular semigroup (although the latter is ambiguous). More generally, in an arbitrary semigroup an element is called ''group-bound'' if it has a power that belongs to a subgroup. Epigroups have applications to ring theory. Many of their properties are studied in this context. Epigroups were first studied by Douglas Munn in 1961, who called them ''pseudoinvertible''. Properties * Epigroups are a generalization of periodic semigroups, thus all finite semigroups are also epigroups. * The class of epigroups also contains al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Monoid
In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can be described by a rational function. Definition Consider a monoid ''M''. Consider a pair (''A'',''L'') where ''A'' is a finite subset of ''M'' that generates ''M'' as a monoid, and ''L'' is a language on ''A'' (that is, a subset of the set of all strings ''A''∗). Let ''φ'' be the map from the free monoid ''A''∗ to ''M'' given by evaluating a string as a product in ''M''. We say that ''L'' is a ''rational cross-section'' if ''φ'' induces a bijection between ''L'' and ''M''. We say that (''A'',''L'') is a ''rational structure'' for ''M'' if in addition the kernel of ''φ'', viewed as a subset of the product monoid ''A''∗×''A''∗ is a rational set. A quasi-rational monoid is one for which ''L'' is a rational relation: a rational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
