An algebraic approach to physical scales

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of "positive space" and its rational powers. Positive spaces are "semi-vector spaces" on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields "scaled spaces" that are suitable to describe spaces with physical dimensions mathematically. We deal also with scales regarded as fields described by sections of "scale bundles". Keywords: semi-vector spaces, scales, units of measurement. MSC 2000: primary 15A69; secondary 12K10, 16Y60, 70Sxx.