Mathematical Theory of Transfer, II

Abstract

In this chapter, we will present our general formalization of transfer. This extends the concept of a τ-automorphism, from the 2-level case, to the n-level case. In the latter case, the transfer automorphism is, itself, a multi-level structure in which transfer is going on within several layers simultaneously. This will become fundamental for example to understanding robotics, where limbs are being transferred simultaneously in several levels, or perceptual organization, which we argue is structured in the same way. The multi-level structure of transfer will be algebraically formalized as a control-nested (i.e., multi-level) τ-automorphism. An ordinary τ-automorphism (Definition 3.2, p. 89) has a control-nested relation with its fiber group. However, a controlnested τ-automorphism has a control-nested hierarchy within itself. Correspondingly the automorphism group is structured in this way. Once again, formalizing transfer in terms of an automorphism group has a number of profound advantages, the most important being the fact that all the transferred copies are contained within one group, together with the algebraic relationships involved.