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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 con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...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] |
Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa. Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order. A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiisomorphism
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets ''A'' and ''B'' is an isomorphism from ''A'' to the opposite of ''B'' (or equivalently from the opposite of ''A'' to ''B''). If there exists an antiisomorphism between two structures, they are said to be ''antiisomorphic.'' Intuitively, to say that two mathematical structures are ''antiisomorphic'' is to say that they are basically opposites of one another. The concept is particularly useful in an algebraic setting, as, for instance, when applied to rings. Simple example Let ''A'' be the binary relation (or directed graph) consisting of elements and binary relation \rightarrow defined as follows: * 1 \rightarrow 2, * 1 \rightarrow 3, * 2 \rightarrow 1. Let ''B'' be the binary relation set consisting of elements and binary relation \Rightarrow defined as follows: * b \Rightarrow a, * c \Rightarrow a, * a \Rightarrow b. Note that the opposite of ''B'' (denoted ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, , p. 19 According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations." Null semigroup Let ''S'' be a semigroup with zero element 0. Then ''S'' is called a ''null semigroup'' if ''xy'' = 0 for all ''x'' and ''y'' in ''S''. Cayley table for a null semigroup Let ''S'' = be (the underlying set of) a null semigroup. Then the Cayley table for ''S'' is as given below: Left zero semigroup A semigroup in which every element i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a branch of mathematics, 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 0. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ... ''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 lan ... [...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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Table
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a groupsuch as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's centercan be discovered from its Cayley table. A simple example of a Cayley table is the one for the group under ordinary multiplication: History Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation ''θ'' ''n'' = 1". In that paper they were referred to simply as tables, and were merely illustrativethey came to be known as Cayley tables later on, in honour of their creator. Structure and layout Because many Cayley tables describe groups that are not abelian, the product ''ab'' with respect to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, , p. 19 According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations." Null semigroup Let ''S'' be a semigroup with zero element 0. Then ''S'' is called a ''null semigroup'' if ''xy'' = 0 for all ''x'' and ''y'' in ''S''. Cayley table for a null semigroup Let ''S'' = be (the underlying set of) a null semigroup. Then the Cayley table for ''S'' is as given below: Left zero semigroup A semigroup in which every element i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
