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Aperiodic Semigroup
In mathematics, an aperiodic semigroup is a semigroup ''S'' such that every element ''x'' ∈ ''S'' is aperiodic, that is, for each ''x'' there exists a positive integer ''n'' such that ''x''''n'' = ''x''''n'' + 1. An aperiodic monoid is an aperiodic semigroup which is a monoid. Finite aperiodic semigroups A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. In terms of Green's relations, a finite semigroup is aperiodic if and only if its ''H''-relation is trivial. These two characterizations extend to group-bound semigroups. A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic.Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," ''Information and Control'', Vol 8 No. 2, pp. 190–194, 1965. A consequence of the Krohn–Rho ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star-free Language
A regular language is said to be star-free if it can be described by a regular expression constructed from the letters of the alphabet, the empty set symbol, all boolean operators – including complementation – and concatenation but no Kleene star.Lawson (2004) p.235 For instance, the language of words over the alphabet \ that do not have consecutive a's can be defined by (\emptyset^c aa \emptyset^c)^c, where X^c denotes the complement of a subset X of \^*. The condition is equivalent to having generalized star height zero. An example of a regular language which is not star-free is \, i.e. the language of strings consisting of an even number of "a". Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids.Lawson (2004) p.262 They can also be characterized logically as languages definable in FO the first-order logic over the natural numbers with the less-than relation, as the counter-free languages and as languages defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monogenic Semigroup
In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups. Structure The monogenic semigroup generated by the singleton set is denoted by \langle a \rangle . The set of elements of \langle a \rangle is . There are two possibilities for the monogenic semigroup \langle a \rangle : * ''a'' ''m'' = ''a'' ''n'' ⇒ ''m'' = ''n''. * There exist ''m'' ≠ ''n'' such that ''a'' ''m'' = ''a'' ''n''. In the former case \langle a \rangle is isomorphic to the semigroup ( , + ) of natural numbers under addition. In such a case, \langle a \rangle is an ''infinite monogenic semigroup'' and the element ''a'' is said to have ''infinite order''. It is sometimes called the ''free monogenic semigroup'' because it is also a free semigroup with one generator. In the latter case let ''m'' be the smallest positive integer such that ''a'' ''m'' = ''a'' ''x'' for some positive integer ''x'' ≠ ''m'', and let ''r'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup With Two Elements
In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements: * O2, the null semigroup of order two, * LO2, the left zero semigroup of order two, * RO2, the right zero semigroup of order two, * (, ∧) (where "∧" is the logical connective "and"), or equivalently the set under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, * (Z2, +2) (where Z2 = and "+2" is "addition modulo 2"), or equivalently (, ⊕) (where "⊕" is the logical connective " xor"), or equivalently the set under multiplication: the only group of order two. The semigroups LO2 and RO2 are antiisomorphic. O2, and are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and are bands. Determination of semigroups with two elements Choosing the set as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup With Three Elements
In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication. Multiplication of integers is associative, and the product of any two of these three integers is again one of these three integers. There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 39 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or antiisomorphic so that there are essentially only 18 possibilities.Andreas DistlerClassification and enumeration of finite semigroups, PhD thesis, University of St. Andrews One of these is C3, the cyclic group with three elements. The others all have a semigroup with two elements as subsemigroups. In the example above, the set under multiplication cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wreath Product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H (sometimes known as the ''bottom'' and ''top''), there exist two variations of the wreath product: the unrestricted wreath product A \text H and the restricted wreath product A \text H. The general form, denoted by A \text_ H or A \text_ H respectively, requires that H acts on some set \Omega; when unspecified, usually \Omega = H (a regular wreath product), though a different \Omega is sometimes implied. The two variations coincide when A, H, and \Omega are all finite. Either variation is also denoted as A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode U+2240). The notion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krohn–Rhodes Theory
In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. In doing their research, though, the authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups. Definitions and description of the Krohn–Rhodes theorem Let ''T'' be a semigroup. A semigroup ''S'' that is a homomorphic image of a subsemigroup of ''T'' is said to be a divisor of ''T''. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup ''S'' is a divisor of a finite alternating wreath product of fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntactic Monoid
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L. Syntactic quotient The free monoid on a given set 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 syntactic equivalence (induced by S). The ''right syntactic equivalence'' is the equivalence relation :s \sim_S t \ \Leftrigh ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', 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 magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automata Theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function, which takes the previous state and current input symbol as its arguments. Automata theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group-bound Semigroup
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 all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
