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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] |
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] |
Right Quotient
In mathematics and computer science, the right quotient (or simply quotient) of a language L_1 with respect to language L_2 is the language consisting of strings ''w'' such that ''wx'' is in L_1 for some string ''x'' in Formally: L_1 / L_2 = \ In other words, for all the strings in L_1 that have a suffix in L_2, the suffix is removed. Similarly, the left quotient of L_1 with respect to L_2 is the language consisting of strings ''w'' such that ''xw'' is in L_1 for some string ''x'' in L_2. Formally: L_2 \backslash L_1 = \ In other words, we take all the strings in L_1 that have a prefix in L_2, and remove this prefix. Note that the operands of \backslash are in reverse order: the first operand is L_2 and L_1 is second. Example Consider L_1 = \ and L_2 = \. Now, if we insert a divider into an element of L_1, the part on the right is in L_2 only if the divider is placed adjacent to a ''b'' (in which case ''i'' ≤ ''n'' and ''j'' = ''n'') or adjacent to a ''c' ... [...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 rationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
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] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Star
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize Automata theory, automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction. Definition Given a set V, define :V^=\ (the set consists only of the empty string), :V^=V, and define recursively the set :V^=\ for each i>0. V^i is called the i-th power of V, it is a shorthand for the Concatenation#Concatenation of sets of strings, concatenation of V by itself i times. That is, ''V^i'' can be understood to be the set of all strings that can be represented as the concatenation of i members from V. The definition of Kleene star on V is : V^*=\bigcup_V^i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
