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Disjunctive Set
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the minimal monoid that recognizes the language L. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism. Syntactic quotient An alphabet is a finite set. The free monoid on a given alphabet is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element. Given a subset S of a free monoid M, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of S by an element m from M is the set :S \ / \ m=\. Similarly, the left quotient is :m \setminus S=\. Syntactic equivalence The syntactic quotient induces an equivalence relation on M, called the syntactic relation, or synt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding Equivalence class, quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Definition The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for group (mathematics), groups, ring (mathematics), rings, vector spaces, module (mathematics), modules, semigroups, lattice (order), lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Trace Monoid
In computer science, a trace is an equivalence class of strings, wherein certain letters in the string are allowed to commute, but others are not. Traces generalize the concept of strings by relaxing the requirement for all the letters to have a definite order, instead allowing for indefinite orderings in which certain reshufflings could take place. In an opposite way, traces generalize the concept of sets with multiplicities by allowing for specifying some incomplete ordering of the letters rather than requiring complete equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier and Dominique Foata in 1969 to give a combinatorial proof of MacMahon's master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization poin ... [...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] |
Dyck Language
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of brackets. The set of Dyck words forms a Dyck language. The simplest, Dyck-1, uses just two matching brackets, e.g. ( and ). Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. Formal definition Let \Sigma = \ be the alphabet consisting of the symbols and Let \Sigma^ denote its Kleene closure. The Dyck language is defined as: : \. Context-free grammar It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal , and the production: : That is, ''S'' is either the empty string () or is " , an element of the Dyck language, the matching ", and an element of the D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicyclic Monoid
A bicyclic molecule () is a molecule that features two joined rings. Bicyclic structures occur widely, for example in many biologically important molecules like α-thujene and camphor. A bicyclic compound can be carbocyclic (all of the ring atoms are carbons), or heterocyclic (the rings' atoms consist of at least two elements), like DABCO. Moreover, the two rings can both be aliphatic (''e.g.'' decalin and norbornane), or can be aromatic (''e.g.'' naphthalene), or a combination of aliphatic and aromatic (''e.g.'' tetralin). Three modes of ring junction are possible for a bicyclic compound: * In spiro compounds, the two rings share only one single atom, the spiro atom, which is usually a quaternary carbon. An example of a spirocyclic compound is the photochromic switch spiropyran. * In fused/condensed bicyclic compounds, two rings share two adjacent atoms. In other words, the rings share one covalent bond, ''i.e.'' the bridgehead atoms are directly connected (''e.g ... [...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] |
Minimal Automaton
In automata theory (a branch of theoretical computer science), DFA minimization is the task of transforming a given deterministic finite automaton (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language. Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. Minimal DFA For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names). The minimal DFA ensures minimal computational cost for tasks such as pattern matching. There are three classes of states that can be removed or merged from the original DFA without affecting the language it accepts. * Unreachable states are the states that are not reachable from the initial state of the DFA, for any input string. These states can be removed. * D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Monoid
In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' called the transition function. Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states ''Q''. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of ''Q''. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors. Transformation semigroups and monoid acts : A transformation semigroup or transformation monoid is a pair (M,Q) consisting of a set ''Q'' (often called the "set of states") and a semigroup or monoid ''M'' of functions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submonoid
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 (category theory), 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 string (computer science), strings built from a given set of Character (computing), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Monoid
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magma (algebra), magmas, where the operation is associative, or as a generalization of group (mathematics), groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutativity, commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
