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


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).

Calendario de la edición:
Volume Open
Temas de la revista:
Mathematics, General Mathematics