topic entry on the algebraic foundations of mathematics

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

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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

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Categ

Defines

universal algebra

Defines

algebraic structure

Defines

logic algebra

Defines

co-algebra

Defines

gebra

Defines

K-algebra

Defines

quantum algebra

Defines

lattice algebra

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