Incomplete idempotent Schröder quasigroups and related packing designs (Q1909661)

A Schröder quasigroup is a quasigroup satisfying the identity \((x\cdot y)\cdot (y\cdot x)= x\). In this paper the authors provide the 7 missing cases from the proof by \textit{C. J. Colbourn} and \textit{D. R. Stinson} [Edge-coloured designs with block size four, Aequationes Math. 36, No. 2-3, 230--245 (1988; Zbl 0658.05016)] to prove that there exists an idempotent Schröder quasigroup of order \(n\) if and only if \(n\) is a positive integer congruent to \(0\) or \(1\) modulo \(4\) and \(n\neq 5\) or \(9\). For \(n\equiv 2, 3\pmod 4\) no such quasigroup can exist, however the next closest thing is a incomplete idempotent Schröder quasigroup of order \(n\) with a hole of side 2 (an \(\text{IISQ}(n, 2)\)). The authors prove that there exists an \(\text{IISQ}(n, 2)\) for any positive integers \(n\equiv 2, 3\pmod 4\), \(n\geq 7\), except for \(n= 10\). The methods used in this paper are standard recursive and direct constructions.