Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2943010.22363/2658-4670-2021-29-4-387-398Research ArticleOn involutive division on monoidsKroytorOleg K.<p>PhD student of Department of Applied Probability and Informatics</p>kroytor_ok@pfur.ruhttps://orcid.org/0000-0002-5691-7331MalykhMikhail D.<p>Doctor of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher in Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research</p>malykh_md@pfur.ruhttps://orcid.org/0000-0001-6541-6603Peoples’ Friendship University of Russia (RUDN University)Joint Institute for Nuclear Research1211202129438739812112021Copyright © 2021, Kroytor O.K., Malykh M.D.2021<p style="text-align: justify;">We consider an arbitrary monoid <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>M</mi><annotation encoding="LaTeX">M</annotation></semantics></math>, on which an involutive division is introduced, and the set of all its finite subsets Set<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>M</mi><annotation encoding="LaTeX">M</annotation></semantics></math>. Division is considered as a mapping <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>:</mo><mi>S</mi><mi>e</mi><mi>t</mi><mi>M</mi><mo></mo><mi>M</mi></mrow><annotation encoding="LaTeX">{d:SetM \times M}</annotation></semantics></math>, whose image <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>(</mo><mi>U</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="LaTeX">{d(U,m)}</annotation></semantics></math> is the set of divisors of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>m</mi><annotation encoding="LaTeX">m</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>U</mi><annotation encoding="LaTeX">U</annotation></semantics></math>. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the Set<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>M</mi><annotation encoding="LaTeX">M</annotation></semantics></math> element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>X</mi><annotation encoding="LaTeX">X</annotation></semantics></math>. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.</p>involutive monomial divisionGröbner basisинволютивное мономиальное делениебазис Грёбнера<p>1. Introduction The creation of the technique of involutive bases as an alternative to the classical Grbner bases and its application to the study of ideals in polynomial and differential rings is undoubtedly one of the most important contributions made by V.P. Gerdt and his disciples in computer algebra. The concept of involutive division came to algebras from the compatibility studies of systems of partial differential equations, dating back to the works of Riquier [1], Janet [2], Thomas [3]. Since the mid-1990s, V.P. Gerdt and his students A.Yu. Zharkov and Yu.A. Blinkov have published a series of papers in which this concept was developed in an abstract algebraic form and indicated the wide possibilities of using involutive bases as an alternative to Kroytor O.K., Malykh M.D., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ the usual Grbner bases. The first example of involutive division - Pomare division - was introduced by Zharkov in 1993 [4]-[7]. In general terms, the concept of involutive division was introduced by Gerdt and Blinkov in [8]-[11]. V.P. Gerdt strove for an axiomatic presentation of the concept of involutive division, especially emphasizing this in his report made at RUDN University in November 2020 [12]. In our opinion, the theory of divisions on monoids, cleared of applied issues, looks like a self-sufficient and very elegant theory, which is complete only to the extent that was of interest for applied researchers. We have tried to present it in general terms. We based on 1.2 from the Dr. Sci. thesis by Blinkov [13], but have significantly revised the terminology. The fact is that the creators of this theory obviously intended to give it a topological interpretation, but, unfortunately, they never did it. Therefore, a number of terms (continuity of division, closure of sets) refer to this so far unknown topology. In our opinion, this topology is the Zariski topology, and therefore incidental analogies taken from the ℝ topology greatly hinder its development. 2. Divisions on monoids Definition 1. A set is called a monoid if a binary associative operation called multiplication is specified on it, and there is an element 1 such that 1</p>[C. Riquier, Les Systèmes d’Equations aux Dérivées Partielles. Paris: Gauthier-Villars, 1910.][M. Janet, “Systèmes d’équations aux dérivées partielles,” Journals de mathématiques, 8e série, vol. 3, pp. 65-151, 1920.][J. Thomas, Differential systems. 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Gerdt, “Gröbner bases and involutive methods for algebraic and differential equations,” Mathematical and computer modelling, vol. 25, no. 8-9, pp. 75-90, 1997. DOI: 10.1016/S0895-7177(97)00060-5.][V. P. Gerdt and Y. A. Blinkov, “Involutive divisions of monomials,” Programming and Computer Software, vol. 24, no. 6, pp. 283-285, 1998.][Y. A. Blinkov, “Division and algorithms in the ideal membership problem [Deleniye i algoritmy v zadache o prinadlezhnosti k idealu],” Izvestija Saratovskogo universiteta, vol. 1, no. 2, pp. 156-167, 2001, in Russian.][V. P. Gerdt. “Compact involutive monomial bases.” (2020), [Online]. Available: https://events.rudn.ru/event/102.][Y. A. Blinkov, “Involutive methods applied to models described by systems of algebraic and differential equations [Involyutivnyye metody issledovaniya modeley, opisyvayemykh sistemami algebraicheskikh i differentsial’nykh uravneniy],” in Russian, Ph.D. dissertation, Saratov State University, Saratov, 2009.][J. Apel, “A Gröbner approach to involutive bases,” Journal of Symbolic Computation, vol. 19, no. 5, pp. 441-458, 1995. DOI: 10.1006/jsco. 1995.1026.][A. Y. Zharkov and Y. A. Blinkov, “Involution approach to solving systems of algebraic equations,” in Proceedings of the 1993 International IMACS Symposium on Symbolic Computation. Laboratoire d’Informatique Fondamentale de Lille, France, 1993, pp. 11-16.]