Quasigroups satisfying Stein's third law with a specified number of idempotents

Quasigroups satisfying Stein's third law (QSTL for short) have been associated with other types of combinatorial configurations, such as cyclic orthogonal arrays. These have been studied quite extensively over the years by various researchers, including Curt Lindner. An idempotent model of a QSTL of order v (briefly QSTL(v)), corresponds to a perfect Mendelsohn design of order v with block size four (briefly a (v,4,1)-PMD) and these are known to exist if and only if v≡0,1(mod4), except for v=4,8. There is a QSTL(4) with two idempotents and it is known that a QSTL(8) contains either 0 or 4 idempotents. In this paper, we formally investigate the existence of a QSTL(v) with a specified number n of idempotent elements, briefly denoted by QSTL(v,n). The necessary conditions for the existence of a QSTL(v,n) are v≡0,1(mod4), 0≤n≤v, and v−n is even. We show that these conditions are also sufficient with few definite exceptions and a handful of possible exceptions. Holey perfect Mendelsohn designs of type 4nu1 with block size four (HPMD(4nu1) for short) are useful to establish the spectrum of QSTL(v,n). In particular, we show that for 0≤u≤8, an HPMD(4nu1) exists if and only if n≥max(4,⌈u/2⌉+1), except possibly (n,u)=(12,1).