Varieties of groupoids and quasigroups generated by linear-bivariate polynomials over ring Z_n

Some varieties of groupoids and quasigroups generated by linear-bivariate polynomials $P(x,y)=a+bx+cy$ over the ring ${\mathbb{Z}}_n$ are studied. Necessary and sufficient conditions for such groupoids and quasigroups to obey identities which involve one, two, three (e.g. Bol-Moufang type) and four variables w.r.t. $a$, $b$ and $c$ are established. Necessary and sufficient conditions for such groupoids and quasigroups to obey some inverse properties w.r.t. $a$, $b$ and $c$ are also established. This class of groupoids and quasigroups are found to belong to some varieties of groupoids and quasigroups such as medial groupoid(quasigroup), F-quasigroup, semi automorphic inverse property groupoid(quasigroup) and automorphic inverse property groupoid(quasigroup).