Abstract
We define a Kulakov algebraic system as a three-sorted algebraic system satisfying the axioms of a physical structure. We prove a strong version of Ionin’s Theorem on the equivalence of the rank \( (2,2) \) physical structure to the structure of an abstract group. We consider nongroup Kulakov algebraic systems and characterize Kulakov algebraic systems over arbitrary groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Along with the term a physical structure the term a phenomenological symmetric geometry of two sets is used, or briefly a geometry of two sets.
Sometimes it is called multibase or heterogeneous.
References
Kulakov Yu. I., Elements of the Theory of Physical Structures, Universe Contract Company, Moscow (2004) [Russian].
Kulakov Yu. I., “A mathematical formulation of the theory of physical structures,” Sib. Math. J., vol. 12, no. 5, 822–824 (1972).
Mikhailichenko G. G., “Solution of functional equations in the theory of physical structures,” Dokl. Akad. Nauk SSSR, vol. 206, no. 5, 1056–1058 (1972).
Bardakov V. G. and Simonov A. A., “Rings and groups of matrices with a nonstandard product,” Sib. Math. J., vol. 54, no. 3, 393–405 (2013).
Kyrov V. A. and Bogdanova R. A., “The groups of motions of some three-dimensional maximal mobility geometries,” Sib. Math. J., vol. 59, no. 2, 323–331 (2018).
Helmholtz H., “On the facts underlying geometry,” in: On Foundations of Geometry, GITTL, Moscow (1956), 366–388.
Mikhailichenko G. G., “Two-dimensional geometries,” Dokl. Akad. Nauk SSSR, vol. 260, no. 4, 803–805 (1981).
Lev V. Kh., “Three-dimensional geometries in the theory of physical structures,” Methodological and Technological Problems of Information-Logical Systems: Computational Systems, vol. 125, 90–104 (1988) [Russian].
Kyrov V. A., “An analytic embedding of some two-dimensional geometries of maximal mobility,” Sib. Electr. Math. Reports, vol. 16, 916–937 (2019).
Vityaev E. E., “Numerical, algebraic, and constructive representation of one physical structure,” Logical Foundations MOZ (Methods for Discovery of Regularities), Computer Systems (Vychisl. Sistemy) [Russian], vol. 107, 40–51 (1985).
Ionin V. K., “Abstract groups as physical structures,” Systemology and Methodological Problems of Information-Logical Systems (Vychisl. Sistemy) [Russian], no. 135, 40–43 (1990).
Simonov A. A., “Pseudomatrix groups and physical structures,” Sib. Math. J., vol. 56, no. 1, 177–190 (2015).
Simonov A. A., “On generalized sharply \( n \)-transitive groups,” Izv. Math., vol. 78, no. 6, 1207–1231 (2014).
Plotkin B. I., Universal Algebra, Algebraic Logic, and Databases, Kluwer, Dordrecht (1994).
Malcev A. I., Algebraic Systems, Springer and Akademie, Berlin, Heidelberg, and New York (1973).
Kurosh A. G., Lectures in General Algebra, Elsevier, Amsterdam (2014).
Kulakov Yu. I., “On a principle lying at the foundation of classical physics,” Dokl. Akad. Nauk SSSR, vol. 193, no. 1, 72–75 (1970).
Simonov A. A., Kulakov Y. I., and Vityaev E. E., “On an algebraic definition of laws,” J. Math. Psychol., vol. 58, 13–20 (2014).
Serovaiskii S. Ya., “Mathematical basis of physical structures theory,” Intern. J. Math. Phys., vol. 4, no. 1, 10–28 (2013).
Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory, Dover, Mineola (2004).
Funding
The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314–2019–0011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 6, pp. 1357–1368. https://doi.org/10.33048/smzh.2021.62.611
Rights and permissions
About this article
Cite this article
Neshchadim, M.V., Simonov, A.A. Kulakov Algebraic Systems on Groups. Sib Math J 62, 1100–1109 (2021). https://doi.org/10.1134/S0037446621060112
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446621060112