Reflection spaces, partial K-loops and K-loops (Q545486)

The points in a reflection space are involutions in some group \(\Gamma\). A reflection space is called \(elliptic\) if there is a point \(p\) such that \(xp=px\) for some point \(x\neq p\). The authors show that in a non-elliptic reflection space any two distinct points have at most one midpoint. Moreover a transitive non-elliptic reflection space \(P\) is a regular permutation set. In order to get similar results for elliptic reflection spaces the authors assume the existence of a midpoint domain; they also introduce a partial \(K\)-loop with fibration.