# PlanetPhysics/Topic Entry on the Algebraic Foundations of Mathematics

This is a new topic on the algebraic foundations of mathematics.
a. Universal (or general) algebra : is defined as the (meta) mathematical study of \htmladdnormallink{general theories {http://planetphysics.us/encyclopedia/GeneralTheory.html} of algebraic structures} rather than the study of specific cases, or models of algebraic structures.
b. Various, specifically selected algebraic structures, such as :
JB- and JL- algebras, Poisson and $C^{*}$ - or C*- algebras,
* sigma-algebra and T-algebras of monads
* K-algebras
* categorical algebra
* F-algebra/coalgebra in category theory
* category of categories as a foundation for mathematics: Functor Categories and 2-category
* super-categories and topological `supercategories'
* 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)
* Index of supercategories
Remark The last items of HDA and SA are more precisely understood in the context of, or as generalizations/ extensions of, universal algebras.
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* ↑ Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168--1182.
* 1 2 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.
* ↑ "Structure in Mathematics and Logic: A Categorical Perspective", Philosophia Mathematica, 3, 209--237.
* ↑ Awodey, S., 2004, "An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism", Philosophia Mathematica, 12, 54--64.
* ↑ Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
* ↑ Baez, J. \& Dolan, J., 1998a, "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes", Advances in Mathematics, 135, 145--206.
* ↑ Baez, J. \& Dolan, J., 2001, "From Finite Sets to Feynman Diagrams", Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.
* ↑ Baez, J., 1997, "An Introduction to n-Categories", Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
* ↑ Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics , 32 : 539-561.
* ↑ 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.
* ↑ Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics , 35 (4), 475--486.
* ↑ Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) -- Systems. Bulletin of Mathematical Biophysics 35 , 213-217.
* ↑ 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.
* ↑ Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology , 39 : 249-258.
* ↑ Baianu, I.C.: 1980a, Natural Transformations of Organismic Structures., Bulletin of Mathematical Biology ,42 : 431-446.
* ↑ Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) --Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report : Failed to parse (syntax error): {\displaystyle \\http://www.ag.uiuc.edu/fs401/QAuto.pdf } and $http://www.medicalupapers.com/quantum+automata+math+categories+baianu/$
* ↑ Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes , 16 Nos. 1-2: 65-122.
* ↑ 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.
* ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
* ↑ Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
* ↑ Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
* ↑ Bell, J. L., 1981, "Category Theory and the Foundations of Mathematics", British Journal for the Philosophy of Science, 32, 349--358.
* ↑ Bell, J. L., 1982, "Categories, Toposes and Sets", Synthese, 51, 3, 293--337.
* ↑ Bell, J. L., 1986, "From Absolute to Local Mathematics", Synthese, 69, 3, 409--426.
* ↑ Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
* ↑ Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
* ↑ Blass, A. and Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111--140.
* ↑ Blass, A. and Scedrov, A., 1992, "Complete Topoi Representing Models of Set Theory", Annals of Pure and Applied Logic , 57, no. 1, 1--26.
* ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
* ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \`{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris. \bibitem (BJk4) Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, \emph{Applied Categorical Structures} 12 : 63-80.
* ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007a, \emph{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
* ↑ 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.
* ↑ 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.
* ↑ Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., \emph{Theory and Applications of Categories} 10 , 71-93.
* ↑ Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr. , 71: 273-286.
* ↑ Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. G\'{e om. Diff.} 17 , 343-362.
* ↑ 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.
* 1 2 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80 : 1-34.
* ↑ Bucur, I., and Deleanu A. (1968). Introduction to the Theory of Categories and Functors . J.Wiley and Sons: London
* ↑ Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179 , 291-317.
* ↑ Bunge, M., 1984, "Toposes in Logic and Logic in Toposes", Topoi, 3, no. 1, 13-22.
* ↑ Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math , \textbf {179}: 291-317.
* ↑ Cartan, H. and Eilenberg, S. 1956. Homological Algebra , Princeton Univ. Press: Pinceton.
* ↑ Cohen, P.M. 1965. Universal Algebra , Harper and Row: New York, London and Tokyo.
* ↑ Connes A 1994. Noncommutative geometry . Academic Press: New York.
* ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.