This article is cited in 7 papers

Quasigroups and their applications

V. A. Artamonov<sup>abc

a</sup> Russian foreign trade academy
^b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^c The Russian Presidential Academy of National Economy and Public Administration

Abstract: A survey of results obtained within the project 0AAAA-A16-116070810025-5 and the recent joint project with Indian algebraists S.Chakrabarti, S. Gangopahyay, S. Pal and also with Russian participants V.T. Markov, A.E. Pankratiev.
The aim of projects is a study of algebraic properties of finite polynomially complete quasigroups, the problem of their recognition from its Latin square and constructions of polynomially complete quasigroups of sufficiently large order. We are also interested in poly nomially complete quasigroups with no subquasigroups. There are found sufficient conditions of polynomial completeness of a quasigroups $Q$ in terms of a group $G(Q)$. For example it suffices if $G(Q)$ acts doubly transitive in $Q$. There is found a behaviour of $G(Q)$ under isotopies.
It is shown that any finite quasigroup can be embedded into a polynomial complete one. The results are applied for securing an information.

Keywords: quasigroups, Latin squres, permutation groups, transitivity.

UDC: 512.57, 512.54

Received: 12.06.2018
Accepted: 17.08.2018

DOI: 10.22405/2226-8383-2018-19-2-111-122

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