Sameness between based universal algebras
نویسندگان
چکیده
The abstract representation-free Algebra of the past century is unable to formalize algebra sameness. A counterexample shows a universal transformation that cannot be representation-free. Then, to define transformations between based universal algebras we must introduce the representations corresponding to the bases, contrary to what was possible in general vector spaces and believed possible in universal algebras. Our universal notion of transformation comes from a triple characterization concerning three representation facets: the determinations of the Menger system, analytic monoid and endomorphism representation corresponding to a base. Hence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformations that were coordinate-free. Universal transformations allow us to check the actual invariance of general algebraic constructions, contrary to the seeming invariance of representation-free thinking. Contrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.
منابع مشابه
Research Summary
Introduction Higher-dimensional algebra is the study of generalizations of algebraic concepts obtained through a process called ‘categorification’. My research in higher-dimensional algebra develops and explores categorified Lie algebras, also called Lie 2-algebras, and their relationships to low-dimensional topology. In the mid-1990’s, Crane [C,CF] coined the term categorification to refer to ...
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