# 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. 1.
2. 2.

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

3. 3.

Quantum logic algebras

4. 4.

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

5. 5.

Algebra over a set

6. 6.

7. 7.

K-algebras

8. 8.
9. 9.

Graphs generated by free groups

10. 10.

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

11. 11.

Hypergraphs generated by free groupoids

12. 12.

Double algebras

13. 13.

Index of algebras

14. 14.
15. 15.

F-algebra/coalgebra in category theory

16. 16.

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

17. 17.

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

18. 18.

super-categories and topological ‘supercategories’

19. 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. 20.

Index of supercategories

21. 21.

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

22. 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

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…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