Commutative hypercomplex numbers and the geometry of two sets

The main task of the theory of phenomenologically symmetric geometries of two sets is the classification of such geometries. In this paper, by complexing with associative hypercomplex numbers, functions of a pair of points of new geometries are found by the functions of a pair of points of some well-known phenomenologically symmetric geometries of two sets (FS GDM). The equations of the groups of motions of these geometries are also found. The phenomenological symmetry of these geometries is established, that is, functional relationships are found between the functions of a pair of points for a certain finite number of arbitrary points. In particular, the $s + 1$-component functions of a pair of points of the same ranks are determined by single-component functions of a pair of points of the FS of GDM ranks $(n,n)$ and $(n + 1,n)$. Finite equations of motion group and equation expressing their phenomenological symmetry are found.