The families of affine semi-algebraic sets over a real-closed field K and semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properies of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation into an algebraic object weaker than a group and culminates with a version of the Jordan-Holder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in K-n is n-dimensional over the scalar ultrafilters. [References: 10]
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