Geometries over universal left conjugacy closed quasifields (Q1357623)

A loop \(L\) is a Bol-loop iff \(\lambda_x \lambda_y \lambda_x= \lambda_{x(yx)} \) holds for all \(x,y\in L\) where \(\lambda_x: L\to L\) is defined by \(\lambda_xy: =xy\). Let \(x \backslash y\) and \(y/x\) be (uniquely) defined by \(x(x\backslash y) =y\) and \((y/x)x =y\) resp., \(L\) is a left conjugacy closed loop (LCC) iff \(\lambda_x^{-1} \lambda_y \lambda_x= \lambda_{x \backslash yx}\). An LCC is universal (ULCC) iff all its isotopic images are also LCC. An equivalent property of ULCC are either of the both identities \[ (rx) (r\backslash (yz)= \bigl[(rx \cdot r \backslash y)/x\bigr] \cdot \bigl[r\backslash (rx \cdot z)\bigr], \] \[ r\bigl( x\backslash (yz) \bigr)= \biggl\{r \bigl[x \backslash (y\cdot r\backslash x) \bigr] \biggr\} \cdot \bigl[x \backslash (rz) \bigr]. \] A quasifield whose multiplicative loop is a ULCC is a ULCC quasifield. The author considers proper ULCC quasifields (i.e. ULCC quasifields with non-associative multiplication) and ULCC planes, i.e. affine planes over such quasifields. He establishes conditions for the property that an André quasifield [cf. e.g. \textit{D. R. Hughes} and \textit{F. C. Piper}, `Projective planes', Springer, New York-Heidelberg-Berlin (1973; Zbl 0267.50018)] is a ULCC quasifield. Especially finite André quasifields are ULCC. He characterizes ULCC planes by configuration conditions.