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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 rationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Group (mathematics), 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 id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene 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] |
Regulator Monoid
Regulator may refer to: Technology * Regulator (automatic control), a device that maintains a designated characteristic, as in: ** Battery regulator ** Pressure regulator ** Diving regulator ** Voltage regulator * Regulator (sewer), a control device used in a combined sewer system * Regulator (timepiece), a device in mechanical timepieces attached to the balance spring for adjusting the rate of the balance wheel * Regulator precision pendulum clock, originally used as a time-standard for adjusting or ''regulating'' other clocks and watches * Regulator, the throttle of a steam engine * Regulator, a component of Uilleann pipes, a form of bagpipes Science * Regulator (mathematics), a positive real number used in Dirichlet's unit theorem * Regulator (biology), an animal that is able to maintain a constant internal environment * Regulator gene, a gene involved in controlling the expression of one or more other genes * Regulator, an auxiliary physics concept used in regularization M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Monoid
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Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Semigroup
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator. Formally, let S be a semigroup and A be a finite set of generators. Then an ''automatic structure'' for S with respect to A consists of a regular language L over A such that every element of S has at least one representative in L and such that for each a \in A \cup \, the relation consisting of pairs (u,v) with ua = v is regular, viewed as a subset of (''A''# × ''A''#)*. Here ''A''# is ''A'' augmented with a padding symbol.. The concept of an automatic semigroup was generalized from automatic groups by Campbell et al. (2001) Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Element
In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation ''zero'' may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous. Definition Formally, let be a set ''S'' with a closed binary operation • on it (known as a magma). A zero element (or an absorbing/annihilating element) is an element ''z'' such that for all ''s'' in ''S'', . This notion can be refined to the notions of left zero, where one requires only that , and right zero, where . Absorbing elements are particularly interesting for semigroups, especially the multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite State Transducer
A finite-state transducer (FST) is a finite-state machine with two memory ''tapes'', following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton (FSA) that maps between two sets of symbols. An FST is more general than an FSA. An FSA defines a formal language by defining a set of accepted strings, while an FST defines a Finitary relation, relation between sets of strings. An FST will read a set of strings on the input tape and generate a set of relations on the output tape. An FST can be thought of as a translator or relater between strings in a set. In morphological parsing, an example would be inputting a string of letters into the FST, the FST would then output a string of morphemes. Overview An automaton can be said to ''recognize'' a string if we view the content of its tape as input. In other words, the automaton computes a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures 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 object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's Theorem
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language ∅ is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
