<abstract xmlns="http://www.w3.org/1999/xhtml">In this study, some linear-bivariate polynomials <italic>P(x, y)</italic> = <italic>a</italic> + <italic>bx</italic> + <italic>cy</italic> that generate quasigroups over the ring Zn are studied. By defining a suitable binary operation * on the set <italic>Hp</italic>(Zn) of all linear-bivariate polynomials of the form <italic>Pf</italic> (x,y) = <italic>fi(a, b, c)</italic> + <italic>f2(a,b,c)x</italic> + <italic>f3(a,b,c)y</italic> where f1, f2, f3 : Zn x Zn x Zn<italic>-></italic> 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 <italic>P(x,</italic> y), then it is shown that (Pp (Zn), *) &lt; (Hp(Zn), *). The group PP (Zn) is found to be isomorphic to the symmetric group <italic>S</italic>3 and to <italic>S</italic>Pp(Zn) &lt; <italic>S</italic>6. A Bol loop of order 36 is constructed using the group PP(Zn). </abstract>

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).

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

Department of Mathematics, Obafemi Awolowo University

Department of Mathematics, Federal University of Agriculture

Annals of West University of Timisoara - Mathematics and Computer Science

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...

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

Published Online: Jan 15, 2013

Page range: 45 - 53

DOI: https://doi.org/10.2478/v10324-012-0014-3

This content is open access.