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Rational Set
In computer science, more precisely in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra. A rational set generalizes the notion of rational (or regular) language (understood as defined by regular expressions) to monoids that are not necessarily free. Definition Let (N,\cdot) be a monoid with identity element e. The set \mathrm(N) of rational subsets of N is the smallest set that contains every finite set and is closed under * union: if A,B\in \mathrm(N) then A\cup B\in \mathrm(N) * product: if A,B\in \mathrm(N) then A\cdot B=\\in\mathrm(N) * Kleene star: if A\in \mathrm(N) then A^*=\bigcup_^\infty A^i \in\mathrm(N) where A^0=\ is the singleton containing the identity element, and where A^=A^n \cdot A. This means that any rational subset of N can be obtained by taking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finitely Generated 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 theory), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Marcel-Paul Schützenberger
Marcel-Paul "Marco" Schützenberger (24 October 1920 – 29 July 1996) was a French mathematician and Doctor of Medicine. He worked in the fields of formal language, combinatorics, and information theory.Herbert Wilf, Dominique Foata, ''et al.'',In Memoriam: Marcel-Paul Schützenberger, 1920-1996," ''Electronic Journal of Combinatorics'', served from University of Pennsylvania Dept. of Mathematics Server, article dated 12 October 1996, retrieved from WWW on 4 November 2006. In addition to his formal results in mathematics, he was "deeply involved in struggle against the votaries of eo-arwinism",Foata, Dominique, "In Memoriam," ''op. cit.'' a stance which has resulted in some mixed reactions from his peers and from critics of his stance on evolution. Several notable theorems and objects in mathematics as well as computer science bear his name (for example Schutzenberger group or the Chomsky–Schützenberger hierarchy). Paul Schützenberger was his great-grandfather. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. As a result of this work, Eilenberg and Mac Lane developed the field of category theory, for which they are now best known. Eilenberg was a member of Bourbaki and, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Jean-Éric Pin
Jean-Éric Pin is a French mathematician and theoretical computer scientist known for his contributions to the algebraic automata theory and semigroup theory. He is a CNRS research director. Biography Pin earned his undergraduate degree from ENS Cachan in 1976 and his doctorate (Doctorat d'état) from the Pierre and Marie Curie University in 1981. Since 1988 he has been a CNRS research director at Paris Diderot University. In the years 1992–2006 he was a professor at École Polytechnique. Pin is a member of the Academia Europaea (2011) and an EATCS fellow (2014). In 2018, Pin became the first recipient of the Salomaa Prize in Automata Theory, Formal Languages, and Related Topics. Notable Work Pin is the author of the prominent textbook ''Varieties of Formal Languages'' on automata theory and formal language theory In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Recognizable Set
In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some homomorphism to a finite monoid. Recognizable sets are useful in automata theory, formal languages and algebra. This notion is different from the notion of recognizable language. Indeed, the term "recognizable" has a different meaning in computability theory. Definition Let N be a monoid, a subset S\subseteq N is recognized by a monoid M if there exists a homomorphism \phi from N to M such that S=\phi^(\phi(S)), and recognizable if it is recognized by some finite monoid. This means that there exists a subset T of M (not necessarily a submonoid of M) such that the image of S is in T and the image of N \setminus S is in M \setminus T. Example Let A be an alphabet: the set A^* of words over A is a monoid, the free monoid on A. The recognizable subsets of A^* are precisely the regular languages. Indeed, such a language is recognized by the transitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Series
In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring, and the indeterminates adjoined are not assumed to commute. They can be regarded as algebraic expressions of a formal language over a finite alphabet. Definition Let ''R'' be a semiring and ''A'' a finite alphabet. A ''non-commutative polynomial'' over ''A'' is a finite formal sum of words over ''A''. They form a semiring R\langle A \rangle. A ''formal series'' is a ''R''-valued function ''c'', on the free monoid ''A''*, which may be written as :\sum_ c(w) w . The set of formal series is denoted R\langle\langle A \rangle\rangle and becomes a semiring under the operations :c+d : w \mapsto c(w) + d(w) :c\cdot d : w \mapsto \sum_ c(u) \cdot d(v) A non-commutative polynomial thus corresponds to a function ''c'' on ''A''* of finite support. In the case when ''R'' is a ring, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Index
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same size as ''H'', the index is related to the orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' is finite, the formula may be written as , G:H, = , G, / ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z. ** A locally cyclic group is a group in which every finitely gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
