Schröder quasigroups with a specified number of idempotents

Schröder quasigroups have been studied quite extensively over the years. Most of the attention has been given to idempotent models, which exist for all the feasible orders vv, where v≡0,1(mod4) except for v=5,9v=5,9. There is no Schröder quasigroup of order 55 and the known Schröder quasigroup of order 99 contains 66 non-idempotent elements. It is known that the number of non-idempotent elements in a Schröder quasigroup must be even and at least four. In this paper, we investigate the existence of Schröder quasigroups of order vv with a specified number kk of idempotent elements, briefly denoted by SQ(v,k)SQ(v,k). The necessary conditions for the existence of SQ(v,k)SQ(v,k) are v≡0,1(mod4), 0≤k≤v0≤k≤v, k≠v−2k≠v−2, and v−kv−k is even. We show that these conditions are also sufficient for all the feasible values of vv and kk with few definite exceptions and a handful of possible exceptions. Our investigation relies on the construction of holey Schröder designs (HSDs) of certain types. Specifically, we have established that there exists an HSD of type 4nu14nu1 for u=1,9u=1,9, and 1212 and n≥max{(u+2)/2,4}n≥max{(u+2)/2,4}. In the process, we are able to provide constructions for a very large variety of non-idempotent Schröder quasigroups of order vv, all of which correspond to v2×4v2×4 orthogonal arrays that have the Klein 44-group as conjugate invariant subgroup.