Geodesic
Hypercomplex numbers in some geometries of two sets. I
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2017), pp. 19-29

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The most important problem in the theory of phenomenologically symmetric geometries of two sets is classification of these geometries. In this work we find metric functions of these new geometries by metric functions of some known phenomenologically symmetric geometries of two sets (PS GTS) with the help of complexification by associative hypercomplex numbers. We also find equations of motion groups of these geometries and phenomenological symmetry of these geometries, i. e., functional relationship between metric functions is specified for definite finite number of arbitrary points. In particular, by single-component metric function of PS GTS of $(2,2)$, $(3,2)$, $(3,3)$ ranks we define $(n+1)$-component metric functions of the same ranks. We find finite equations of motion group and equation expressing their phenomenological symmetry.
Keywords: geometry of two sets, phenomenological symmetry, group symmetry, hyper-complex numbers. 
@article{IVM_2017_7_a2,
     author = {G. G. Mikhailichenko and V. A. Kyrov},
     title = {Hypercomplex numbers in some geometries of two sets. {I}},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {19--29},
     publisher = {mathdoc},
     number = {7},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_7_a2/}
}
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G. G. Mikhailichenko; V. A. Kyrov. Hypercomplex numbers in some geometries of two sets. I. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2017), pp. 19-29. http://geodesic.mathdoc.fr/item/IVM_2017_7_a2/