\(2\)-perfect \(m\)-cycle systems can be equationally defined for \(m=3\), \(5\), and \(7\) only
A natural quasigroup associated with a 2-perfect \(m\)-cycle system is examined, and it is shown that except when the cycle length \(m\) is 3, 5, or 7, the 2-perfect \(m\)-cycle systems cannot be equationally defined. The proof proceeds by producing a quasigroup associated with a 2-perfect \(m\)-cycle system, for which a particular homomorphic image does not arise from a 2-perfect \(m\)-cycle system.
scientific article; zbMATH DE number 840229