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# Topic: Magma (algebra)

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 Magma (algebra) - Wikipedia, the free encyclopedia In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object). en.wikipedia.org /wiki/Magma_(algebra)   (572 words)

 Magma Magma is molten rock often located inside a magma chamber[?] beneath the surface of the earth. Magma in Earth is a complex high-temperature silicate solution that is ancestral to all igneous rocks, both intrusive and effusive. Magma is under high pressure and sometimes emerges through volcanic vents in the form of lava. www.ebroadcast.com.au /lookup/encyclopedia/ma/Magma.html   (96 words)

 Magma computer algebra system - Wikipedia, the free encyclopedia Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. Magma is produced and distributed by the Computational Algebra Group within the School of Mathematics and Statistics of the University of Sydney. en.wikipedia.org /wiki/Magma_computer_algebra_system   (190 words)

 PREFACE The computer algebra system Magma is designed to provide a software environment for computing with the structures which arise in areas such as algebra, number theory, algebraic geometry and (algebraic) combinatorics. Algebraic structures are first classified by variety: a variety being a class of structures having the same set of defining operators and satisfying a common set of axioms. Magma comprises a novel user programming language based on the principles outlined above together with program code and databases designed to support computational research in those areas of mathematics which are algebraic in nature. www.math.lsu.edu /magma/preface.htm   (713 words)

 PREFACE   (Site not responding. Last check: 2007-11-06) The algebra system Magma is designed to provide a software environment for computing with the structures which arise in algebra, geometry, number theory and (algebraic) combinatorics. Algebraic structures are first classified by variety, that is, a class of structures having the same set of defining operators and satisfying a common set of axioms. The Magma implementation of the Dixon-Schneider algorithm for computing the table of ordinary characters of a finite group is based on an earlier version written for Cayley by Gerhard Schneider (Karlsruhe). www.math.uga.edu /~matthews/DOCS/MAGMA/preface.html   (1528 words)

 [No title]   (Site not responding. Last check: 2007-11-06) In the case of an algebra defined over a field \$K\$, quotient algebras, composition series, the composition factors and the Jacobson radical may be constructed. Magma takes 280 seconds to determine that the curve y^2 + x*y = x^3 - 215*x + 1192 has group Z_2 x Z x Z, and 100 seconds to determine that the curve y^2 = x^3 + 36861504658225*x^2 + 1807580157674409809510400*x has rank 13. In accordance with the Magma philosophy, a graph may be studied under the action of an automorphism group. www.symbolicnet.org /ftpsoftware/magmaa2.2.txt   (3296 words)

 Vector Enumeration The algebra may be the group algebra of an fp-group, in which case the resulting module will be a matrix representation of the group, or it might be a more general fp-algebra, such as a Hecke algebra or a quotient of a polynomial ring. The permutation module of degree 4 of this algebra is presented by 1 generator (as it is transitive) and the submodule generator b - 1. This is represented as a table, with columns indexed by the generators of the algebra, and rows indexed by the basis of the space. www.math.ufl.edu /help/magma/text469.html   (1546 words)

 [ref] 63 Magma Rings free algebras and free associative algebras, with or without one, where the magma is a free magma or a free semigroup, or a free magma-with-one or a free monoid, respectively. Note that a free Lie algebra is not a magma ring, because of the additional relations given by the Jacobi identity; see Magma Rings modulo Relations for a generalization of magma rings that covers such structures. Note that in a magma ring, the addition of elements is in general different from an addition that may be defined already for the elements of the magma; for example, the addition in the group ring of a matrix group does in general not coincide with the addition of matrices. www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/htm/ref/CHAP063.htm   (1273 words)

 Magma Computer Algebra   (Site not responding. Last check: 2007-11-06) Magma is a relatively new system for computer algebra based and developed at the University of Sydney, Australia, by the computational algebra group, headed by Dr John Cannon. A Magma workshop at the University of the Western Cape, Cape Town, South Africa, organized to coincide with the 39th South-African Mathematical Society conference, November 4--6, 1996. This was part of the inaugural Computational Algebra and Number Theory conference (CANT90) at the University of Sydney, held at the occassion of the retirement of Professor Tim Wall, from November 6 to 9 of 1990. staff.science.uva.nl /~wieb/magma   (298 words)

 Construction of Group Algebras and their Elements This representation allows group algebras to be defined over any group in which the elements have a canonical form, including potentially infinite matrix groups over a ring of characteristic 0 or even free groups. Note however, that operations in such algebras are limited, as the length of the representing arrays may grow exponentially with the number of multiplications performed. We first construct the default group algebra A = R[G] where R is the ring of integers and G is the symmetric group on three points. www.umich.edu /~gpcc/scs/magma/text926.htm   (695 words)

 Introduction   (Site not responding. Last check: 2007-11-06) The free magma on X is the set of the x_i along with all bracketed expressions in the x_i, e.g., ((x_1, x_2), ((x_1, x_3), x_2)). In Magma we do not work with bases of the free Lie algebra, as they are not of much use for our main problem: to find a basis and a multiplication table for a finitely-presented Lie algebra. This means that, mathematically speaking, in Magma the algebra that we call the free Lie algebra, is in fact the free nonassociative anticommutative algebra. wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1121.htm   (422 words)

 Construction of Subalgebras, Ideals and Quotient Algebras A subalgebra or ideal of a group algebra A is not in general a group algebra; hence, a different type is needed for these objects. Its elements are elements of the group algebra and it remains closely connected to its defining group algebra. The minimal ideals are the simple Wedderburn components of the group algebra, corresponding to the absolutely irreducible characters of the group, and their dimensions are the squares of the character degrees. www.math.wisc.edu /help/magma/text582.html   (869 words)

 Magma (algebra) at opensource encyclopedia   (Site not responding. Last check: 2007-11-06) In abstract algebra, a magma is a particularly basic kind of algebraic structure. Specifically, a magma consists of a set X with a single binary operation on it. It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of X, with operation the joining of trees at the root. www.wiki.tatet.com /Magma_(algebra).html   (372 words)

 Magma Computational Algebra System Home Page Magma is a large, well-supported software package designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics. March 2006: The Magma group will be holding a two-day workshop on computational number theory at Sydney University on March 21-22; interested people are welcome to attend. October 18, 2004: The Algebraic Geometry and Number Theory with Magma conference was held October 4 - 8, 2004 at the Institute Henri Poincaré, Paris. magma.maths.usyd.edu.au /magma   (268 words)

 Computer Algebra Systems The design of Magma emphasizes structural computation, that is, the ability to construct canonical representations of structures, thereby permitting such operations as membership testing, determination of structural properties and isomorphism testing. Magma has been developed by the Computational Algebra Group, headed by Dr J. Cannon, in the School of Mathematics and Statistics at the University of Sydney. Though its scope for symbolic and algebraic computation is very limited at this moment, its performance of doing several major algebraic operations in the polynomial ring is considerably high to cope with practical problems. www.informatik.uni-leipzig.de /~graebe/ComputerAlgebra/Systems/overview.html   (1833 words)

 SAL- Mathematics - Computer Algebra Systems The major purpose of a Computer Algebra System (CAS) is to manipulate a formula symbolically using the computer. Macaulay 2 -- algebraic geometry and commutative algebra. MAGMA -- a system for algebra, number theory, geometry and combinatorics. www.sai.msu.su /sal/A/1/index.shtml   (538 words)

 Overview   (Site not responding. Last check: 2007-11-06) Although vector spaces are, of course, subsumed under general modules, we present a separate treatment of them, firstly because of their importance and secondly because their theory is somewhat cleaner than that of a general module. Magma users who are unfamiliar with the language of module theory will find a self-contained treatment of the vector space machinery in Chapter 30. In the Magma universe, rectangular matrices are regarded as forming a module (actually a bimodule). www.math.uiuc.edu /Software/magma/text383.html   (192 words)

 [No title] In Magma V2.5, the field K may be taken to be either a finite field, Q or a number field. An algebraic function field is a finite degree extension of a rational function field K(x). A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one. www.umich.edu /~gpcc/scs/magma/text73.htm   (2563 words)

 [No title]   (Site not responding. Last check: 2007-11-06) The conference is intended to inform algorithm designers and users of recent developments in cognate areas of computational algebra, number theory, and geometry and inform key theoretical mathematicians of the possibilities offered by the computational algebra tools that are now becoming available. Algebra studies basic questions about discrete structure and symmetry in the world. Algebraic invariants are used to answer biochemists' questions about knotting in DNA and anthropologists have even used algebraic classifications to organize native artifacts in the southwestern U.S. For many years mathematicians have studied algebra with little more than their imaginations and a pencil and paper. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9626327.txt   (470 words)

 Mathematics applications available Magma is a radically new system designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics. Magma provides a mathematically rigorous environment for computing with algebraic and geometric objects. The strongest point of the package is the inclusion of truly novel routines for doing "cylindrical algebraic decomposition", a way of solving systems of polynomial inequalities in several variables over the real numbers. www.math.uiuc.edu /consult/use_system/apps/mathsw.html   (1588 words)

 ACKNOWLEDGEMENTS The Magma facilities for groups and monoids defined by confluent rewrite systems, as well as automatic groups, are supported by this code. Eamonn's extensions in Magma of this package for generating p-groups, computing automorphism groups of p-groups, and deciding isomorphism of p-groups are also included. The Magma implementation of the Dixon--Schneider algorithm for computing the table of ordinary characters of a finite group is based on an earlier version written for Cayley by Gerhard Schneider (Karlsruhe). www.math.lsu.edu /magma/ackn.htm   (4222 words)

 About "The Magma Computational Algebra System"   (Site not responding. Last check: 2007-11-06) A system designed by the Computational Algebra Group, headed by Dr J. Cannon, in the School of Mathematics and Statistics at the University of Sydney, to solve computationally hard problems in algebra, number theory, geometry and combinatorics. The design of Magma emphasizes structural computation - the ability to construct canonical representations of structures, thereby permitting such operations as membership testing, determination of structural properties and isomorphism testing. The Math Forum is a research and educational enterprise of the Drexel School of Education. mathforum.org /library/more_info.html?id=2524   (107 words)

 COHOMOLOGY OF 2-GROUPS Thanks are due to John Cannon and Allan Steel of the MAGMA project for numerous instances of help with the tools to make the programs work and for their enthusiastic support. The first is all groups, modules and linear algebra, The second stage is all Gröbner basis and commutative algebra calculations. The intersections of the kernels of restriction to the maximal subgroups is known as the essential cohomology. www.math.uga.edu /~jfc/groups/cohomology.html   (3660 words)

 Release Notes V2.7 (June 30, 2000)   (Site not responding. Last check: 2007-11-06) It is of course not possible to construct explicitly the closure of a field, but the system works by automatically constructing larger and larger algebraic extensions of an original base field as needed during a computation, thus giving the illusion of computing in the algebraic closure of the base field. An algebraic function field F/k (in one variable) over a field k is a field extension F of k such that F is a finite field extension of k(x) for an element x in F which is transcendental over k. An affine algebra in Magma is simply the quotient ring of a multivariate polynomial ring P = K[x_1,...,x_n] over a field K by an ideal J of P. Such rings arise commonly in commutative algebra and algebraic geometry. www.sci.kuniv.edu.kw /magma/text74.html   (6112 words)

 [No title] The object of This course is to show you how to use Magma to explore some of the topics in the Algebra 3 class 1.1 - How to get help Magma has an extensive help facility. The situation in Magma is slightly different, since different types of groups require different algorithms. We can ask Magma this: */ IsFinite(H); // We can also find the order of H #H; // There is another way of doing this c,d := IsFinite(H); c,d; /* Since the group H is finite, every element of H has finite order, i.e., some power of it will be the identity element. www.math.bgu.ac.il /~bessera/magma/lecture1   (936 words)

 Annenberg Media Exhibits -- Volcanoes - Melting Rocks When a body of magma rises through the denser rock layers toward Earth's surface, some of it remains liquid. Magma that has reached the surface is called lava. Because rocks are made up of collections of minerals that melt at different temperatures, the makeup of the rock being melted affects the magma that results. www.learner.org /exhibits/volcanoes/meltrock.html   (208 words)

 gcvr: algebra software   (Site not responding. Last check: 2007-11-06) Besides general computer algebra software, the Geometry Center maintains a several specialized computer algebra packages packages that actually focus on mathematical algebra. Magma is a programming language designed for the investigation of algebraic and combinatorial structures, together with a vast collection of library functions and databases for implementing standard algebraic operations. Historically, Magma is essentially a superset of Cayley, with greatly expanded capability for general algebraic objects. www.geom.uiuc.edu /local/overview/mathcomp/alg.html   (287 words)

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