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# Topic: Universal algebra

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 Universal algebra Info - Encyclopedia WikiWhat.com   (Site not responding. Last check: 2007-11-07) Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. The motivation for the field is the many examples of algebras (in the sense of universal algebra), such as monoids, rings, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these fields, but with universal algebra, you can prove them once and for all for every kind of algebraic system. www.wikiwhat.com /encyclopedia/u/un/universal_algebra.html   (689 words)

 Variety (universal algebra) - Wikipedia, the free encyclopedia In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Those equations are statements from the predicate calculus involving universal quantifiers and equality only: each is a mathematical identity enforced in each model, for example the commutative law, or the absorption law. It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations, so the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational. en.wikipedia.org /wiki/Variety_(universal_algebra)   (683 words)

 PlanetMath: Hopf algebra   (Site not responding. Last check: 2007-11-07) The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. This is version 9 of Hopf algebra, born on 2002-10-18, modified 2005-07-12. planetmath.org /encyclopedia/HopfAlgebra.html   (267 words)

 Whitehead’s Early Philosophy of Mathematics In Universal Algebra Whitehead sought to achieve what he calls generality by trying to unify by a common interpretation apparently disparate algebraic systems that to many did not appear to be mathematics at all. Universal Algebra, in precisely this sense, is a poor framework for mathematics insofar as it unites spatial manifolds and symbolic logic by introducing the common notion of an algebraic manifold (Whitehead’s terminology) or a semi-group (current standard terminology), an object with very little structure or intrinsic interest. While Universal Algebra does have its moments, it is rich mathematically only insofar as Whitehead transcends the generality of his algebraic manifolds and deals in the specifics of Boolean algebra or Grassmannian manifolds. www.religion-online.org /showarticle.asp?title=2850   (7542 words)

 08: General algebraic systems The appeal of the subject in its early years was probably due to its universality, but the work of a few dozen people during the past two decades has added a dimension of depth to the breadth that was the original trademark of universal algebra. "Algebra" is a very broad section of mathematics; there are separate index pages here for specific algebraic categories (groups, fields, etc.) This heading focuses both on the broad principles covering all of algebra and on specific algebraic constructs not included in those other areas. Universal algebra is arguably more a topic in Logic (03C05) (Model Theory), hence there is significant overlap. www.math.niu.edu /~rusin/known-math/index/08-XX.html   (510 words)

 PlanetMath: universal enveloping algebra   (Site not responding. Last check: 2007-11-07) ) is isomorphic to the skew polynomial algebra Cross-references: polynomial, Lie bracket, left ideal, maximal, irreducible, theory, representation, generates, clear, injective, map, Poincaré-Birkhoff-Witt theorem, isomorphic, universal property, two-sided ideal, vector space, generated by, tensor algebra, algebra, commutator, structure, homomorphism, unity, associative, field, Lie algebra This is version 4 of universal enveloping algebra, born on 2002-09-18, modified 2006-03-22. planetmath.org /encyclopedia/UniversalEnvelopingAlgebra.html   (190 words)

 A Sketch of the Prehistory of Universal Algebra   (Site not responding. Last check: 2007-11-07) A Sketch of the Prehistory of Universal Algebra In the case of Universal Algebra what suggests itself as the beginning, at least in a narrow sense, is Garrett Birkhoff's paper [1] On the Structure of Abstract Algebras which appeared in the Proceedings of the Cambridge Philosophical Society in 1935. Universal Algebra [he says] is the name applied to that calculus which symbolizes general operations, defined later, which are called Addition and Multiplication. www.maths.utas.edu.au /People/dfs/Papers/GrassmannUAlgpaper/node2.html   (846 words)

 Facts about universal enveloping algebra   (Site not responding. Last check: 2007-11-07) In mathematics, the universal enveloping algebra construction of abstract algebra is applied to a Lie algebra L in order to pass from a non-associative structure to a more familiar and associative algebra over a field U(L) while preserving the representation theory. Starting with the tensor algebra T(L) on the vector space underlying L, we should make U(L) be the quotient of T(L) made by imposing the relations like a'b-b'a = [a,b] for a and b in (the image in T(L)) of L, where on the RHS the bracket now means the given Lie algebra product. For example if L is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order. www.supercrawler.com /Facts/universal_enveloping_algebra.html   (490 words)

 20th WCP: The Model Theory Of Dedekind Algebras Each Dedekind algebra is associated with a cardinal value function called the confirmation signature which counts the number of configurations in each isomorphism type occurring in the decomposition of the algebra. A subalgebra, A, of the Dedekind algebra B is a small subalgebra of B provided the cardinality of the domain of A is strictly smaller than the cardinality of the domain of B. Let B be an infinite Dedekind algebra. Since the class of uncountable homogeneous-universal Dedekind algebras is a finitary class, for each such algebra there is a subset of its theory all of whose models of the same cardinality are isomorphic. www.bu.edu /wcp/Papers/Logi/LogiWeav.htm   (3006 words)

 [No title] Differential graded modules, algebras, coalgebras, and Hopf algebras are shortened to dgm, dga, dgc, and dgh, respectively; a comprehensive treatment of these objects is given in [3]. The graded abelian Lie algebra on {xj}, denoted Lab(xj), is the free graded module on the basis {xj}, with the trivial Lie bracket. Er be the composition of algebra isomorphisms H(mr) H(ffr-1) (sWr) -- - ! www.math.purdue.edu /research/atopology/ScottJA/ls-bss.txt   (3926 words)

 [No title] In Grätzer's book "Universal Algebra" three entire chapters are devoted to free algebras; their definition is not very intuitive and requires the previous reading of three other chapters (if only for the notation). Ehr, no. In a universal algebra of a given type, no axioms are of the form "existence." For example, to truly define groups as universal algebras, you do not define them as a set with one operation and the usual axioms (associtivity, existence of neutral elements, existence of inverses). The "free algebra construct" is meant to be the adjoint of this functor, in the sense that it should be a functor F:Sets=>T, with the property that for every set X and every algebra A in T, there exists a natural bijection between the morphism sets: Mor_{Sets}(X,U(A)) and Mor_{T}(F(X),A). www.math.niu.edu /~rusin/known-math/00_incoming/free_alg   (3067 words)

 Taylor & Francis Online Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications. www.crcpress.com /shopping_cart/products/product_detail.asp?sku=C2549   (392 words)

 Universal Algebra   (Site not responding. Last check: 2007-11-07) Hermann Grassmann and the Prehistory of Universal Algebra I was brought to Universal Algebra against my will, as it were by Hermann Grassmann, and the main point of this paper is to describe a piece of Grassmann's work and to ask those who know the subject better than I do whether it may be said to anticipate Universal Algebra. A universal algebra is a set G together with a system of n-ary operations for G; here n may vary and the number of operations may be infinite. www.maths.utas.edu.au /People/dfs/Papers/GrassmannUAlgpaper/node1.html   (398 words)

 Elementary Universal Algebra and Computer Science   (Site not responding. Last check: 2007-11-07) Universal algebra is a branch of mathematics which was originally concerned with abstract generalizations of algebraic concepts. It is perhaps surprising then that the ideas of universal algebra have played a significant role in such areas of computer science as the specification of data types, the semantics of programming languages, and the theory of compilers. Equally surprising is that the seemingly abstract fundamental constructs of universal algebra can be easily implemented on a computer using a symbolic computation system such as Mathematica in a manner that makes it possible to construct and explore interesting computer science examples. www.ii.uib.no /~fredrikm/seminar/wagner.html   (167 words)

 Background   (Site not responding. Last check: 2007-11-07) The universal enveloping algebra U(L) of L is the associative algebra with identity, generated by n symbols which are also denoted by x_1,..., x_n. If the Lie algebra L happens to be (split) semisimple and of characteristic 0, then the universal enveloping algebra has a nice basis described by [Kos66]. In the universal enveloping algebra we use the divided powers y_i^((n)) = (y_i^n/n!), x_i^((n)) = (x_i^n/n!), and the binomials (h_i choose k) = (h_i(h_i - 1)... www.math.lsu.edu /magma/text1103.htm   (293 words)

 [No title] Universal logic stands in the same position with regards to the multiplicity of logics as universal algebra with the multiplicity of algebras. The terminology «universal logic» shows clearly that universal logic is different from universal algebra (and in particular not part of it), but at the same time shows also the spiritual connection. Universal logic can give a new direction to the philosophy of logic, because it provides via modern mathematics, rigour and abstraction, without which philosophy of logic is only metaphorical discussion, bad poetry in the sense of Carnap. www.sorites.org /Issue_12/beziau.htm   (10029 words)

 Universal Algebra for Computer Science   (Site not responding. Last check: 2007-11-07) As the name suggests, Universal Algebra is concerned with the properties of general algebraic systems. Universal Algebra is the name applied to that calculus which symbolizes general operations, defined later, which are called Addition and Multiplication. Constructs of Universal Algebra are at the basis of many areas of Computer Science, among which are Formal Methods, Type Theory, Specification, Complexity Theory, Computer algebra, Uncertainty Management and many more. www.cosc.brocku.ca /~duentsch/teaching/univalg/ua_intro.html   (232 words)

 Universal Algebra and Diagrammatic Reasoning   (Site not responding. Last check: 2007-11-07) Since the introduction of category theory, the old subject of "universal algebra" has diversified into a large collection of frameworks for describing algebraic structures. Our treatment of algebraic theories and PROPs explains how the latter are related to Feynman diagrams, and leads up to an adjunction between algebraic theories and PROPs which is analogous to the relation between classical and quantum physics. Stanley Burris and H.P. Sankappanavar, A Course in Universal Algebra math.ucr.edu /home/baez/universal   (235 words)

 AMCA: Weak Congruences in Universal Algebra by Andreja Tepavcevic Weak congruences were introduced in universal algebra together with other compatible relations generalizing the concept of a congruence (implicitly by F. Sik and his Ph.D. student T.D. Mai in 1974, and under the present name by Vojvodi\'c and Seselja in 1988). The aim of this text is to present basic properties of weak congruences, particularly of lattices of these, and to highlight their applications in universal algebra. L. Is there an algebra such that its weak congruence lattice is isomorphic with L, the diagonal relation being the image of a under the isomorphism. at.yorku.ca /c/a/j/l/04.htm   (659 words)

 Matthew Gould Publications "On extensions of Schreier's theorem to universal algebras," Studia Scientiarum Mathematicarum Hungarica, 1 (1996), pp. "On versatile monoids of endomorphisms," Proceedings of the October 1969 Conference on Universal Algebra, Queens University, Kingston, Ontario (1970), pp. "Subalgebra maps induced by endomorphisms and automorphisms of algebras," Algebra Universalis, 2 (1972), pp. sitemason.vanderbilt.edu /page/g6CFG0   (606 words)

 Evolution of Geometric Algebra and Calculus The idea of geometric algebra was given its modern form and reinvigorated by more than a century of advances in mathematics and physics since Grassmann. The roles of theoretical physics and the Lecture Notes of Marcel Riesz [1958] in stimulating the initial synthesis are described in the article Clifford Algebra and the Interpretation of Quantum Mechanics. The book Space-Time Algebra provided a kind of "proof of concept." It showed how geometric algebra provides compact, coordinate-free formulations for basic equations of physics as well as new insights into their geometric structure. modelingnts.la.asu.edu /html/evolution.html   (745 words)

 LC '98 abstract: Raimon Elgueta   (Site not responding. Last check: 2007-11-07) B. Neumann [2] proved in 1962 that those algebras which are free in some class with $\omega$ free generators are obtained by factorizing the absolutely free algebra with $\omega$ free generators by a fully invariant congruence. If A is an algebra of type L, we define the set of A-structures in K, denoted by K(A), as the class of members of K on the algebra A. We define unions and intersections of members of K(A) in the obvious way, i.e., we join and meet the corresponding relations, respectively. Let X be a class of L-algebras and K a class of L-structures whose underlying algebras are in X. We say that K is a fully invariant system on X if for every h:A-->B with A,B in X, and every M in K(B), the inverse image of M by h is in K(A). www.math.cas.cz /~lc98/abstracts/Elgueta.html   (560 words)

 [No title]   (Site not responding. Last check: 2007-11-07) Algebraic techniques have become a necessary ingredient in most of the exciting recent results in model theory. In the last fifteen years universal algebra has acquired two powerful new tools: commutator theory and tame congruence theory. This project will support a special year in model theory and universal algebra to be held at the University of Illinois at Chicago in 1991-92. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9115761.txt   (164 words)

 About "Hermann Grassmann and the Prehistory of Universal Algebra"   (Site not responding. Last check: 2007-11-07) A definition of universal algebra, with a sketch of its prehistory. Here the term 'family of algebras of a given species' is used in a technical sense meaning a class closed under taking subalgebras, homomorphic images and direct products - or what is nowadays called a variety. The Math Forum is a research and educational enterprise of the Drexel School of Education. mathforum.org /library/view/3253.html   (150 words)

 Universal Algebra and Logic Seminars We show that these systems are equivalent and that they constitute equivalent algebraic semantics for the class of non-associative residuated lattices. We will show that for algebraic lattices, PCC is equivalent to the property that the meet of the pseudo-prime elements is zero. Abstract: We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty. sitemason.vanderbilt.edu /page/f0Cdhu   (2086 words)

 Process Algebra   (Site not responding. Last check: 2007-11-07) Its tools are algebraical languages for the specification of processes and the formulation of statements about them, together with calculi for the verification of these statements. A process algebra was a structure in the sense of universal algebra that satisfied a particular set of axioms. In this meaning the phrase was sometimes used to refer to their own algebraic approach to the study of concurrent processes [BK86b], and sometimes to such algebraic approaches in general [BK86c]. theory.stanford.edu /~rvg/process.html   (352 words)

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