## Abstract

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)* + *f _{2}(a,b,c)x* +

*f*where f

_{3}(a,b,c)y_{1}, f

_{2}, f

_{3}: 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

*S*

_{3}and to

*S*

_{Pp}(

_{Zn}) <

*S*

_{6}. A Bol loop of order 36 is constructed using the group P

_{P}(Z

_{n}).

## Comments (1)

Nice work

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