On single laws for varieties of groupoids associated with strongly \(2\)-perfect \(m\)-cycle systems. (Q1771889)

Let \(K_v\) be the complete graph on \(v\) vertices and let \(S\) denote the set of all vertices. An \(m\)-cycle system \((S,C)\) of order \(v\) is a decomposition of \(K_v\) into the set \(C\) of edge-disjoint cycles of length \(m\). Let \(c\) be a cycle of length \(m\) and \(c(i)\) denote the graph formed from \(c\) by joining all vertices at distance \(i\) in \(c\). If \((S,C)\) is an \(m\)-cycle system of \(K_v\) such that \((S,\{c(i)\mid c\in C\})\) is also a cycle system of \(K_v\) then \((S,C)\) is called an \(i\)-perfect \(m\)-cycle system. For an \(m\)-cycle system \((S,C)\) a binary operation \(*\) on \(S\) can be defined as follows: (1) \(x*x=x\), (2) if \(x\neq y\) then \(x*y=z\) and \(y*x=w\) if and only if \((\dots, w,x,y,z,\dots )\in C\). Then \((S,*)\) is a groupoid associated with \((S,C)\). The authors derive single laws for certain varieties of groupoids associated with strongly \(2\)-perfect \(m\)-cycle systems. These groupoids are quasigroups for \(m = 3,5\) and \(7\) only.