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 Diskr. Mat., 2017, Volume 29, Issue 3, Pages 92–113 (Mi dm1455)

Elementary transformations of systems of equations over quasigroups and generalized identities

S. V. Polin

Academy of Cryptography of Russian Federation

Abstract: The paper is devoted to the study of equations with the left-hand side having the form of a composition of operations which belong to given sets ${{\mathbf{S}}_1},\ldots,{{\mathbf{S}}_n},\ldots$ of quasigroup operations. Elementary transformations are described which allow reducing systems of this kind to the form where all equations except one do not depend essentially on the variable ${x_n}$. A class of systems is said to be Gaussian if every system obtained via such transformations also belongs to this class. It is evident that for Gaussian classes of systems of equations there is an efficient solving algorithm. This motivates the problem of finding conditions under which the class is Gaussian. In this work it is shown that for a class of systems to be Gaussian the operations in the sets ${{\mathbf{S}}_i}$ should satisfy the generalized distributivity law. Sets of operations obeying this condition are to be investigated in the future.

Keywords: systems of equations, quasigroups, Gaussian algorithm, generalized law of distributivity

DOI: https://doi.org/10.4213/dm1455

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English version:
Discrete Mathematics and Applications, 2019, 29:6, 383–399

Bibliographic databases:

UDC: 512.543.7+512.548.7

Citation: S. V. Polin, “Elementary transformations of systems of equations over quasigroups and generalized identities”, Diskr. Mat., 29:3 (2017), 92–113; Discrete Math. Appl., 29:6 (2019), 383–399

Citation in format AMSBIB
\Bibitem{Pol17} \by S.~V.~Polin \paper Elementary transformations of systems of equations over quasigroups and generalized identities \jour Diskr. Mat. \yr 2017 \vol 29 \issue 3 \pages 92--113 \mathnet{http://mi.mathnet.ru/dm1455} \crossref{https://doi.org/10.4213/dm1455} \elib{http://elibrary.ru/item.asp?id=29887804} \transl \jour Discrete Math. Appl. \yr 2019 \vol 29 \issue 6 \pages 383--399 \crossref{https://doi.org/10.1515/dma-2019-0037} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000504837800005} 

• http://mi.mathnet.ru/eng/dm1455
• https://doi.org/10.4213/dm1455
• http://mi.mathnet.ru/eng/dm/v29/i3/p92

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This publication is cited in the following articles:
1. S. V. Polin, “Neobkhodimye usloviya primenimosti algoritma Gaussa k sistemam uravnenii nad kvazigruppami”, Diskret. matem., 30:1 (2018), 95–113
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