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 :


 * 1) Boolean algebra


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


 * 1) quantum logic algebras

JB- and JL- algebras, Poisson and $$C^*$$ - or C*- algebras,
 * 1) quantum operator algebras ( such as : involution, *-algebras, or $$*$$-algebras, von Neumann 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 $$C^*$$-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 $$C^*$$ -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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