, 3 November 2011 (UTC)

Whoa.  I really lost you here.  The affine space was only an analogy.  I do not want to construct something new, only show that there are two classes in common usage: Clifford ring (my name) and Clifford algebra; furthermore, that some inaccuracy occurs because they are not adequately distinguished.  All I'm saying is that there exists a class of objects (Clifford ring) in which no specific subspace is necessarily known as the generating vector space V.  You can hide the identity of the subspace V (and the grading) by "mysterious relabelling".  But here is the point: this relabelling truly does hide which of the basis vectors span V; it is possible to form a product for every element in the ring, but impossible to identify the original subspace V used in the construction.  The quaternions are a ring isomorphic to Cℓ0,2(ℝ), which has a 1,2,1 grading structure.  Yet, given ℍ, can you find a unique decomposition with this grading structure?  No.  In a CA, grading specifically refers to a unique/natural decomposition into (I think it is into a direct sum of) 0-vectors, 1-vectors, 2-vectors etc.  talkcontr 19:22