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Sameness between based universal algebras

Ricci G.

Demonstratio Mathematica

2009

Vol. 42, nr 1

3-22

This is the continuation of the paper "Transformations between Menger systems". To define when two universal algebras with bases "are the same", here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of the Menger system, analytic monoid and endomorphism representation corresponding to a basis. 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. They propose a new interpretation of free algebras as superpositions of "analytic spaces" and deny that our algebras differ from vector spaces at fundamental stages. Contrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

Transformations between Menger system

Ricci G.

Demonstratio Mathematica

2008

Vol. 41, nr 4

743-762

To define transformations between based universal algebras we must introduce representations that depend on the bases, contrary to what was possible for general vector spaces and believed possible for universal algebras. In fact, a counterex-ample shows that by representation-free transformations alone one cannot even ascertain whether a universal algebra has any dimension or not. A transformation notion, which can do, concerns basis dependent Menger systems. It enjoys a basic geometric property of universal algebras, the preservation of reference flocks, and generalizes the transformation groups of Linear Algebra into groupoids.