Schrِder 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 v , where v ≡ 0 , 1 ( mod 4 ) except for v = 5 , 9 . There is no Schröder quasigroup of order 5 and the known Schröder quasigroup of order 9 contains 6 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 v with a specified number k of idempotent elements, briefly denoted by SQ ( v , k ) . The necessary conditions for the existence of SQ ( v , k ) are v ≡ 0 , 1 ( mod 4 ) , 0 ≤ k ≤ v , k ≠ v − 2 , and v − k is even. We show that these conditions are also sufficient for all the feasible values of v and k 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 4 n u 1 for u = 1 , 9 , and 12 and n ≥ m a x { ( 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 v , all of which correspond to v 2 × 4 orthogonal arrays that have the Klein 4 -group as conjugate invariant subgroup.