Some (2, n) combinatorial Stein groupoids.

A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identity x( xy)= yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities. [ABSTRACT FROM AUTHOR]