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Annals of West University of Timisoara - Mathematics and Computer Science

The Journal of West University of Timisoara

Editor-in-Chief: Sasu, Bogdan

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On a Group of Linear-Bivariate Polynomials that Generate Quasigroups over the Ring ℤn

T. G. Jaiyéọlá / E. Ilojide
Published Online: 2013-01-15 | DOI: https://doi.org/10.2478/v10324-012-0014-3


In this study, some linear-bivariate polynomials P(x, y) = a + bx + cy that generate quasigroups over the ring Zn are studied. By defining a suitable binary operation * on the set Hp(Zn) of all linear-bivariate polynomials of the form Pf (x,y) = fi(a, b, c) + f2(a,b,c)x + f3(a,b,c)y where f1, f2, f3 : Zn x Zn x Zn-> Zn, it is proved that (Hp(Zn), *) is a monoid. Necessary and sufficient conditions for it to be a group and abelian group are established. If PP(Zn) is the set of the linear-bivariate polynomials that generate the quasigroups that are the parastro- phes of the quasigroup generated by P(x, y), then it is shown that (Pp (Zn), *) < (Hp(Zn), *). The group PP (Zn) is found to be isomorphic to the symmetric group S3 and to SPp(Zn) < S6. A Bol loop of order 36 is constructed using the group PP(Zn).

: Keywords Quasigroups; parastrophes; linear-bivariate polynomials.

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About the article

Published Online: 2013-01-15

Published in Print: 2012-12-01

Citation Information: Annals of West University of Timisoara - Mathematics, Volume 50, Issue 2, Pages 45–53, ISSN (Print) 1841-3293, DOI: https://doi.org/10.2478/v10324-012-0014-3.

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  • Nice work

    posted by: Temitope Jaiyeola on 2013-09-12 03:57 AM (America/New_York)