An algebraic approach to physical scales (Q970514)

The authors propose the category of positive spaces (i.e. essentially the \(1\)-dimensional semi-vector spaces over \(\mathbb R^{+}\)) and \(q\)-rational maps (for all \(q \in Q\)) between them, as a convenient mathematical setting to formulate the intuitive notions of physical scales and units of measurement. To this end, semi-tensor products, spaces of morphisms and rational powers of a positive space are treated. Besides, using methods of Differential Geometry, the bundles of positive spaces based on spacetime and their semi-linear connections are discussed. A brief review of the relationship to dimensional analysis is also given [e.g., see \textit{G. I. Barenblatt}, Scaling. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (2003; Zbl 1094.00006)].