Algebraic characterization of self-orientation systems

Abstract

In this paper, we try to precise what should be relevant algebraic, topological and axiomatic properties of self-orientation systems. Thanks to a formal apparatus one expects to measure the complexity of self-orientation computation process, to gain accuracy and ultimately to find new algorithms as corollaries of this modeling attempt. The main issue can be depicted as follows : assume that candidate self-orientations form a set S or a vector space V or more generally an algebraic structure A (e.g. a partially ordered set P, a monoid M, a semi-group S,…); if the computation of a self-orientation refers to a motivated choice among possible orientations inside an algebraic structure, one should be able in this structure to separate “good” and “bad” self-orientations. That is, consistency is required and as such needs to be defined. Take a cognitive entity e; a “good” valuation is called a e-model and is noted e┬ and a “bad” valuation is called e-counter-model and is noted e┴. Therefore, consistent self-orientations can be called formal actions provided that the algebraic structure must agree with the separable property which in terms of polynomial algebra corresponds to the reducible property. This paper try to connect algebraic and geometrical representations of actions and axiomatic consistent representations using deductive systems and Hopf Algebras.