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  • Author: T.G. Jaiyeola x
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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 S 3 and to S Pp(Zn) < S 6. A Bol loop of order 36 is constructed using the group PP(Zn).