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Principal Factor
In algebra, the principal factor of a \mathcal-class ''J'' of a semigroup ''S'' is equal to ''J'' if ''J'' is the kernel of ''S'', and to J \cup \ otherwise. Properties * A principal factor is a simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ..., 0-simple or null semigroup.Grillet (1995), p. 50, Proposition 4.9. References Further reading *. * . *. Semigroup theory {{abstract-algebra-stub ... [...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 (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 magmas, where the operation is associative, or as a generalization of 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 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 commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (algebra)
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the se ... [...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 A. H. Clifford and G. B. 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 whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clarendon Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 400 years, OUP has focused primarily on the publication of pedagogic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcel Dekker
Marcel Dekker was a journal and encyclopedia publishing company with editorial boards found in New York City. Dekker encyclopedias are now published by CRC Press, part of the Taylor and Francis publishing group. History Initially a textbook publisher, the company added journal publishing in the 1970s, and encyclopedia publishing in the early 1980s. Serving mathematics, it published a series of ''Lecture Notes in Pure and Applied Mathematics''. The company was purchased by Taylor and Francis in 2003. At that time, it published 78 journals and 300 new books annually. The imprint closed in 2005. As of 2008, they have a total of 26 encyclopedias available. These encyclopedias deal with scientific issues such as agricult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
