Extractions: Vector spaces, say over the rationals, algebraically closed fields of a given characteristic, and free groups are three examples of classes of mathematical structures which are categorical in all uncountable cardinals, i.e. any two uncountable such structures of the same size are isomorphic. In each of these instances, there is a notion of dimension (linear dimension, transcendence degree, the number of generators) which captures the isomorphism-type of the structure. This is a general model-theoretic phenomenon: Th: (Lessmann) If K is a reasonable class of mathematical structures which is categorical in some uncountable cardinal, then inside each mathematical structure there is a pregeometry whose dimension determines the isomorphism-type of the mathematical structure, and furthermore, the class is categorical in all uncountable cardinals. By reasonable, we mean (1) axiomatised using at most countably many first order axioms (the first two examples above, in this case this is the classical Baldwin-Lachlan theorem), or, more generally, (2) axiomatised using not necessarily first order axioms but in such a way that there is a good notion of universal domain (a homogeneous model as in the example of free groups, or a full model). The difficulty in (2) is that the compactness theorem fails. The use of dimension theory to understand mathematical structures works beyond can categoricity: Inside any mathematical structure, we can define what we mean by "A is independent from B (over C)" using the automorphism group of the structure. This independence relation has good properties under very general model-theoretic circumstances, called simplicity and stability (shown by Buechler-Lessmann). Examples of stable and simple mathematical structures are those described above (in each case the independence relation becomes the familiar one: in vector spaces it becomes linear independence, and alebraic independence in an algebraically closed field), as well as Hilbert spaces, where the independence relation coincides with orthogonality.
Logic And Foundations Of Programming At QMW Department of Computer Science, logic and Foundations of Programming group. Members, research projects.Category Science Math Institutions Europe United Kingdom type theory and its semantics; operational semantics; foundations of logic programming.Semantics Universal algebra; category theory; categorical model theory http://www.dcs.qmw.ac.uk/research/theory/
Extractions: Programming The LFP group is one of six research groups here in the Department of Computer Science at Queen Mary University of London The group organises joint theory seminars together with the Theory and Formal Methods Group at Imperial College. We meet for lunch on Wednesdays at one O'clock in room CS/446 above the Student Union shop. Visitors are welcome to attend and join in with our informal presentations and discussions of our current research. There's an incomplete list of lunches. The Department is the site for Hypatia , a directory of research workers in Computer Science and Pure Mathematics and a library of their papers. This document has sections on the group's People Projects and Research David Pym pym@dcs.qmw.ac.uk ), now at Bath Samin Ishtiaq si@dcs.qmw.ac.uk ), now at ARM Ltd Paul Blain Levy pbl@dcs.qmw.ac.uk
Information And Computation -- 1995 A process algebra for timed systems. GL McColm. Pebble games and subroutines inleast fixed point logic. A categorical linear framework for Petri nets. http://theory.lcs.mit.edu/~iandc/ic95.html
Extractions: Michiel Smid . Dynamic rectangular point location with an application to the closest pair problem. Information and Computation , 116(1):1-9, January 1995. Abstract, References, and Citations. BibTeX entry Michael Huth . A maximal monoidal closed category of distributive algebraic domains. Information and Computation , 116(1):10-25, January 1995. Abstract, References, and Citations. BibTeX entry Juanito Camilleri and Glynn Winskel . CCS with priority choice. Information and Computation , 116(1):26-37, January 1995. Abstract, References, and Citations. BibTeX entry Bent Thomsen . A theory of higher order communicating systems. Information and Computation , 116(1):38-57, January 1995. Abstract, References, and Citations. BibTeX entry Anil Nerode Raymond T. Ng , and V. S. Subrahmanian . Computing circumscriptive databases: I. Theory and algorithms. Information and Computation , 116(1):58-80, January 1995. Abstract, References, and Citations. BibTeX entry Naoki Abe . Characterizing PAC-learnability of semilinear sets. Information and Computation , 116(1):81-102, January 1995.
Cours Functoriality. Naturality. Monoidal categories. categorical logic. Hopf algebra.Linear logic. MAT3341, Applied Linear algebra Vector and matrix norms. http://aix1.uottawa.ca/~epaqu045/cours.html
Extractions: Vector and matrix norms. Schur canonical form, QR, LU, Cholesky and singular value decomposition, generalized inverses, Jordan form, Cayley-Hamilton theorem, matrix analysis and matrix exponentials, eigenvalue estimation and the Greshgorin Circle Theorem; quadratic forms, Rayleigh and minima principles. The theoretical and numerical aspects will be studied.
FOM: The Categorical Approach To Logic The context was my point that algebra is not the same as logic, and that the algebraicor categorical approach to logic omits a lot of important information http://www.cs.nyu.edu/pipermail/fom/1998-April/001837.html
Extractions: Wed, 1 Apr 1998 10:39:56 -0500 (EST) Previous message: FOM: history and f.o.m. Next message: FOM: the categorical approach to logic Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: history and f.o.m. Next message: FOM: the categorical approach to logic Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Gnist.no: Fagbokhandelen På Internett Rosebrugh, R. Sets for Mathematics categorical algebra is used the study of geometry,analysis, and algebra. Analysis and logic Articles from four researchers http://www.gnist.no/kategori.php?kategori=PBC
A Reflective Module Algebra With Applications To The Maude Language structured theories in a categorical way and giving some useful results for them.As a first step towards a generic module algebra in a framework logic such as http://maude.cs.uiuc.edu/papers/abstract/Dmodalg_1999.html
Extractions: It has long been recognized that large specifications are unmanageable unless they are built in a structured fashion from smaller specifications using specification-building operations. Modularity and module composition are central notions for specification languages and declarative programming languages. Although Parnas is the author of the possibly earliest work on software modules, Burstall and Goguen were the first to study the semantics of modular specifications and their composition operations in their language Clear. They proposed the idea of ``putting theories together'' by composing them through operations having a clean and logic-independent categorical semantics. Continuing in this line of work, Burstall and Goguen captured the minimal requirements that a logic must meet to be a reasonable specification framework and introduced the notion of institution Categorical techniques have allowed since then the study of specification-building operations with independence of any specific formalism by different authors, giving rise to a large body of research. Algebraic specification is now a mature field of Computer Science because of its mathematical foundations. After Clear, the theory of algebraic specification has been implemented in many computing systems, such as OBJ, ACT ONE, ASL, ASF, PLUSS, LPG, Larch, CASL, etc., and has become an important technique in software engineering methodologies.
Category Theory F. Borceux, Handbook of categorical algebra 13. GM Kelly, Basic Concepts ofEnriched Category Theory. Topos Theory. categorical logic and Type Theory. http://www.kyoto-su.ac.jp/~hxm/categorical/ct/
Logic - CASE Bibliography On Rigid Structures. Journal of Symbolic logic, vol. Eilenberg, S., Harrison, DK,Mac Lane, S., and Roehrl, H., editors, categorical algebra, pages 120. http://david_hewins.tripod.com/papers/id13.html
Extractions: Atzeni, Paolo and Peter P. Chen (1981). Completeness of Query Languages for the Entity-Relationship Model. Entity-Relationship Approach to Information Modeling and Analysis, ed. P.P. Chen. ER Institute, 1981. Republished in Second International Conference on the Entity-Relationship Approach, 1981 (1983). North-Holland.
The Unwritten Book logic. Introduction To Boolean algebra; Dialog logic Felscher'sRules; The Definition Of logic, Sequents and categorical algebra; http://www.uwm.edu/~whopkins/logic/
5 De Morgan's Life And Work Figure 2 De Morgan's notation for the categorical forms AOEI. Besides his workin algebra and logic, De Morgan contributed 712 articles to the ``Penny http://www.hf.uio.no/filosofi/njpl/vol2no1/history/node5.html
Extractions: Next: 6 Boole's Life Up: A Brief History of Previous: 4 British Mathematics in Augustus de Morgan was born as fifth child on the 27th of June 1806 in Madura, India, where his father worked as an officer for the East India Company. His family soon moved to England, where they lived first at Worcester and then at Taunton. His early education was in private schools, where he enjoyed a classic education in Latin, Greek, Hebrew, and mathematics. In 1823, at the age of 16, he entered Trinity College in Cambridge, where the work of the ``Analytical Society'' had already changed the students' schedule so that De Morgan also studied Continental mathematics. In 1826, he graduated as a fourth Wrangler and turned his back on mathematics to study to be a lawyer at Lincoln's Inn in London. But only a year later he revised this decision and applied for a position as professor of mathematics at the newly established University College in London. At the age of 22, with no publications, he was appointed. The work that De Morgan produced in the years to come spanned a wide variety of subjects with an emphasis on algebra and logic. But surprisingly he was not able to connect them. An important work of his was the ``Elements of Arithmetic'', published in 1830, containing a simple yet thorough philosophical treatment of the ideas of number and magnitude. In a paper from 1838 he formally described the concept of mathematical induction and in 1849 in ``Trigonometry and Double Algebra'' he gave a geometrical interpretation of complex numbers.
Bilgi Mathematics Faculty: Prof. Oleg Belegradek 4, 37. On almost categorical theories, Sibirsk. Finitely approximable associativealgebra with unsolvable word problem, algebra and Logika 39, no. logic 65, no http://math.bilgi.edu.tr/people/belegradek/
Category Theory dimensional categories; applications of category theory to algebra, geometry and scientificknowledge that make use of categorical methods logic and Philosophy. http://lgxserver.uniba.it/lei/logica/lgcat_th.htm
Extractions: "The journal Theory and Applications of Categories will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.". Il Journal distribuito gratuitamente via WWW/ftp, dopo essersi registrati. Related Fields SWIF
BIBLIOGRAPHY About DESCENT And CATEGORY THEORY! A. Obtu \book The logic of categories of partial book Introduction to categories,homological algebra, and sheaf Von Eye , et al \book categorical Variables in http://north.ecc.edu/alsani/catbib.html
Extractions: BIBLIOGRAPHY about DESCENT THEORY from W. Tholen home page. Monades et Descente Selected Topics in Algebra An Outline of a Theory of Higher Dimensional Descent The Theory of Descent Triples and Descent An Extension of the Galois Theory of Grothendieck Theory of Categories over a Base Topos Descent Theory for Toposes Effective Descent Morphisms and Effective Equivalence Relations Introduction to Affine Group Schemes BIBLIOGRAPHY about CATEGORY THEORY F. W. Lawvere publications: http://www.acsu.buffalo.edu/~wlawvere Back to Descent and Category Theory WebPage
Extractions: Tomasz Kowalski (JAIST, Japan) In computable algebra and model theory computable isomorphism types of structures have been studied intensively over almost three decades. These include a number of natural classes of structures, such as Boolean algebras, Abelian groups, and lattices. The Handbook of Recursive Mathematics is a good source of results in the area. Here we present two results about computable isomorphisms of Boolean algebras with operators (BAOs). A computable BAO A is one whose domain is a computable subset of N, and whose Boolean operations and the operators are computable functions. If a BAO
Sources On The Philosophy Of Mathematics Mac Lane and Moerdijk, Sheaves in Geometry and logic , SpringerVerlag, 1992.Borceux, Handbook of categorical algebra (three volumes), Cambridge, 1994. http://www.rbjones.com/rbjpub/philos/maths/faq008.htm
Extractions: Several good collections of philosophical papers are available, e.g. and For material on logicism see the writings of Gottlöb Frege and those of Bertrand Russell Some good sources on intuitionism and constructive mathematics are Errett Bishop Douglas Bridges and Michael Beeson . For philosophical material relating to intuitionism Dummett must be mentioned (not withstanding his conspicuous absence from my bibliography). The "anti-foundationalist" heresy may be found in the works of Imre Lakatos Reuben Hersh and Phillip Davis and is copiously represented in . (see Are Foundations Necessary? for my affirmation of orthodoxy, though aimed a computer scientists rather than anti-foundationalists) On the more technical side provides a collection of important papers published in the first three decades of the 20th century. While predominantly by logicians this collection is a valuable resource for philosophers of mathematics. Each paper is prefaced by a contemporary overview and evaluation. There is a lot of material on the foundations of mathematics in the references above.
Listings Of The World Science Math Algebra Category Theory Post Review This site contains online books and research papers on the subjects ofcategorical algebra, categorical logic, categorical geometry, lattice theory http://listingsworld.com/Science/Math/Algebra/Category_Theory/
Bibliography Of G.Rosolini logic, 55, 1990. and Comput., 79, 1988. Rosolini, G. Representation theoremsfor special pcategories, In categorical algebra and its Applications, Ed. http://www.disi.unige.it/person/RosoliniG/biblio.html
Extractions: Power, A. J., Rosolini, G. Fixpoint operators for domain equations Theoret. Comput. Sci. Carboni, A., Rosolini, G., Walters, R.F., editors Theory Appl. Categ. Robinson, E.P., Rosolini, G. An abstract look at realizability , In Computer Science Logic '01 , Ed. L. Fribourg , Lectures Notes in Computer Science, Fiore, M., Rosolini, G. Domains in H Theoret. Comput. Sci. Carboni, A., Rosolini, G. Locally cartesian closed exact completions J.Pure Appl. Alg. Rosolini, G. Equilogical spaces and filter spaces Rend. Circ. Mat. Palermo Rosolini, G. A note on Cauchy completeness for preorders Riv. Mat. Univ. Parma Rosolini, G. Sheaves Spring School on CATEGORICAL METHODS in LOGIC and COMPUTER SCIENCE Birkdedal, L., van Oosten, J., Rosolini, G., Scott, D. S., editors Workshop on Realizability Semantics and Applications , Elsevier Science, Electr. Notes in Theo. Comp. Sci., 1999 Rosolini, G., Streicher, Th. Comparing models of higher type computation , In Workshop on Realizability Semantics and Applications , Ed. Birkdedal, L., van Oosten, J., Rosolini, G., Scott, D. S., Elsevier Science, Electr. Notes in Theo. Comp. Sci., 1999 Birkedal, L., Carboni, A., Rosolini, G., Scott, D. S.
