Commutative Hypercomplex Numbers and the Geometry of Two Sets
The main taskof the theoryof phenomenologically symmetric geometriesoftwo setsis the classiﬁcationof such geometries. In this paper,bycomplexing with associativehypercomplexnumbers, functionsofapairofpointsof new geometries are foundbythe functionsofapairofpointsof somewellknown 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.