Algebraic classification of physical structures with zero. I

An algebraic version is considered of the axiomatization of physical structures. An arbitrary set $R$ with a distinguished element $O$ (zero) is taken as a set of measurements. Under an additional condition, understood to be an analog of the requirement that a physical structure is one-metric, the structure of a topological skew field with zero $O$ is introduced on $R$; and on the object sets $\mathcal M$ and $\mathcal N$, the structure of finite-dimensional vector spaces over the skew field is introduced. This leads to a complete classification of the corresponding physical structures. The classification theorem can be considered also as a variant of the axiomatics connected with a bilinear form on a pair of finite-dimensional vector spaces over a skew field; i.e., the variant which uses, as axioms, only the combinatorial properties of a bilinear form as a map $\mathcal M\times\mathcal N\to R$ (i.e., without the axioms of addition and multiplication).