PlanetPhysics/Topic Entry on the Algebraic Foundations of Mathematics

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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 :

  1. Boolean algebra
  1. Logic lattice algebras or many-valued (MV) logic algebras
  1. quantum logic algebras
  1. quantum operator algebras ( such as : involution, *-algebras, or -algebras, von Neumann algebras,

JB- and JL- algebras, Poisson and - or C*- algebras,

  1. Algebra over a set
  1. sigma-algebra and T-algebras of monads
  2. K-algebras
  1. group algebras
  1. graphs generated by free groups
  1. groupoid algebras and Groupoid -convolution algebras
  1. hypergraphs generated by free groupoids
  1. Double algebras
  1. Index of algebras
  1. categorical algebra
  2. F-algebra/coalgebra in category theory
  3. category of categories as a foundation for mathematics: Functor Categories and 2-category
  1. Index of category theory
  1. super-categories and topological `supercategories'
  2. higher dimensional algebras (HDA) --such as: algebroids, double algebroids, categorical algebroids, double groupoid convolution algebroids, groupoid -convolution algebroids, etc., and Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (ETAS)
  3. Index of supercategories
  1. Index of categories
  1. 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.

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