topic entry on the algebraic foundations of mathematics

This is a contributed topic on the algebraic foundations of mathematics. This topic of algebraic foundations in mathematics will cover a wide range of concepts and areas of mathematics, ranging from universal algebras, algebraic topology to algebraic geometry, number theory and logic algebras.

a. Universal (or general) algebra : is defined as the (meta) mathematical study of general theories of algebraic structures rather than the study of specific cases, or models of algebraic structures.

b. Various, specifically selected algebraic structures, such as :

1.

Boolean algebra
2.

Logic lattice algebras or many-valued (MV) logic algebras
3.

Quantum logic algebras
4.

Quantum operator algebras ( such as : involution, *-algebras, or $*$ -algebras, von Neumann algebras, JB- and JL- algebras, Poisson and $C^{*}$ - or C*- algebras,
5.

Algebra over a set
6.

Sigma-algebra and T-algebras of monads
7.

K-algebras
8.

Group algebras
9.

Graphs generated by free groups
10.

Groupoid algebras and Groupoid $C^{*}$ -convolution algebras
11.

Hypergraphs generated by free groupoids
12.

Double algebras
13.

Index of algebras
14.

Categorical algebra
15.

F-algebra/coalgebra in category theory
16.

Category of categories as a foundation for mathematics: Functor Categories (http://planetmath.org/FunctorCategories) and 2-category (http://planetmath.org/2Category)
17.

Index of category theory (http://planetmath.org/IndexOfCategoryTheory)
18.

super-categories and topological ‘supercategories’
19.

Higher dimensional algebras (HDA) –such as: algebroids, double algebroids, categorical algebroids, double groupoid convolution algebroids, groupoid $C^{*}$ -convolution algebroids, etc., and Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (ETAS)
20.

Index of supercategories
21.

Index of categories (http://planetmath.org/IndexOfCategories)
22.

Index of HDA

Remark The last items of HDA and SA are more precisely understood in the context of, or as generalizations/ extensions of, universal algebras.

References

1 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
2 Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France, 84: 307–317.
3 Auslander, M. 1965. Coherent Functors. Proc. Conf. Cat. Algebra, La Jolla, 189–231.
4 Awodey, S. & Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168–1182.
5 Awodey, S. & Reck, E. R., 2002, Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic., History and Philosophy of Logic, 23, 1, 1–30.
6 Awodey, S. & Reck, E. R., 2002, “Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics”, History and Philosophy of Logic, 23, 2, 77–94.
7 “Structure in Mathematics and Logic: A Categorical Perspective”, Philosophia Mathematica, 3, 209–237.
8 Awodey, S., 2004, “An Answer to Hellman’s Question: Does Category Theory Provide a Framework for Mathematical Structuralism”, Philosophia Mathematica, 12, 54–64.
9 Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
10 Baez, J. & Dolan, J., 1998a, “Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes”, Advances in Mathematics, 135, 145–206.
11 Baez, J. & Dolan, J., 2001, “From Finite Sets to Feynman Diagrams”, Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.
12 Baez, J., 1997, “An Introduction to n-Categories”, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1–33.
13 Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.
14 Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1–4, 1971, Bucharest.
15 Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics, 35(4), 475–486.
16 Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) – Systems. Bulletin of Mathematical Biophysics 35, 213-217.
17 Baianu, I.C. and M. Marinescu: 1974, On A Functorial Construction of (M,R)– Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.
18 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
19 Baianu, I.C.: 1980a, Natural Transformations of Organismic Structures., Bulletin of Mathematical Biology,42: 431-446.
20 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report: $\\ http://www.ag.uiuc.edu/fs401/QAuto.pdf$ and $http://www.medicalupapers.com/quantum+automata+math+categories+baianu/$
21 Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1-2: 65-122.
22 Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.
23 Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
24 Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
25 Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
26 Bell, J. L., 1981, “Category Theory and the Foundations of Mathematics”, British Journal for the Philosophy of Science, 32, 349–358.
27 Bell, J. L., 1982, “Categories, Toposes and Sets”, Synthese, 51, 3, 293–337.
28 Bell, J. L., 1986, “From Absolute to Local Mathematics”, Synthese, 69, 3, 409–426.
29 Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
30 Birkoff, G. & Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
31 Blass, A. and Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111–140.
32 Blass, A. and Scedrov, A., 1992, ”Complete Topoi Representing Models of Set Theory”, Annals of Pure and Applied Logic , 57, no. 1, 1–26.
33 Borceux, F.: 1994, Handbook of Categorical Algebra, vols: 1–3, in Encyclopedia of Mathematics and its Applications 50 to 52, Cambridge University Press.
34 Bourbaki, N. 1961 and 1964: Algèbre commutative., in Èléments de Mathématique., Chs. 1–6., Hermann: Paris.
35 BJk4) Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12: 63-80.
36 Brown, R., Higgins, P. J. and R. Sivera,: 2007a, Non-Abelian Algebraic Topology, in preparation.
http://www.bangor.ac.uk/ mas010/nonab-a-t.html ;
http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf
37 Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., Axiomathes (17): 321–379.
38 Brown R. and T. Porter: 2003, Category theory and higher dimensional algebra: potential descriptive tools in neuroscience, In: Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1, 80-92.
39 Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., Theory and Applications of Categories 10, 71-93.
40 Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr., 71: 273-286.
41 Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. Géom. Diff. 17, 343-362.
42 Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math., 2: 25–61.
43 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80: 1-34.
44 Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. 69: 66-74.
45 Bucur, I., and Deleanu A. (1968). Introduction to the Theory of Categories and Functors. J.Wiley and Sons: London
46 Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179, 291-317.
47 Bunge, M., 1984, ”Toposes in Logic and Logic in Toposes”, Topoi, 3, no. 1, 13-22.
48 Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math, 179: 291-317.
49 Cartan, H. and Eilenberg, S. 1956. Homological Algebra, Princeton Univ. Press: Pinceton.
50 Cohen, P.M. 1965. Universal Algebra, Harper and Row: New York, London and Tokyo.
51 Connes A 1994. Noncommutative geometry. Academic Press: New York.
52 Croisot, R. and Lesieur, L. 1963. Algèbre noethérienne non-commutative., Gauthier-Villard: Paris.

…more to come

Title	topic entry on the algebraic foundations of mathematics
Canonical name	TopicEntryOnTheAlgebraicFoundationsOfMathematics
Date of creation	2013-03-22 18:14:02
Last modified on	2013-03-22 18:14:02
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	35
Author	bci1 (20947)
Entry type	Topic
Classification	msc 08A99
Classification	msc 08A70
Classification	msc 18E05
Classification	msc 18-00
Classification	msc 03-00
Classification	msc 08A05
Synonym	Algebraic Foundations of Mathematics
Related topic	Algebras
Related topic	Graph
Related topic	Hypergraph
Related topic	TopicEntryOnAlgebra
Related topic	IndexOfCategoryTheory
Related topic	NonAbelianStructures
Related topic	JordanBanachAndJordanLieAlgebras
Related topic	AbelianCategory
Related topic	AxiomsForAnAbelianCategory
Related topic	GeneralizedVanKampenTheoremsHigherDimensional
Related topic	AxiomaticTheoryOfSupercategories
Related topic	Categ
Defines	universal algebra
Defines	algebraic structure
Defines	logic algebra
Defines	co-algebra
Defines	gebra
Defines	K-algebra
Defines	quantum algebra
Defines	lattice algebra