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 Z_n are studied. By defining a suitable binary operation * on the set Hp(Z_n) 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 Z_n-> Z_n, it is proved that (H_p(Z_n), *) is a monoid. Necessary and sufficient conditions for it to be a group and abelian group are established. If P_P(Z_n) 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 (P_p (Z_n), *) < (H_p(Z_n), *). The group P_P (Z_n) is found to be isomorphic to the symmetric group S₃ and to S_Pp(_Zn) < S₆. A Bol loop of order 36 is constructed using the group P_P(Z_n).