Pure Latin directed triple systems

It is well-known that, given a Steiner triple system, a quasigroup can be formed by defining an operation · by the identities x · x = x and x ·y = z where z is the third point in the block containing the pair {x, y}. The same is true for a Mendelsohn triple system where the pair (x, y) is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist and we call these Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law x · (y · x) = (x · y) · x but those associated with Latin directed triple systems need not. A directed triple system is said to be pure if when considered as a twofold triple system it contains no repeated blocks. In a previous paper, [Discrete Math. 312 (2012), 597– 607], we studied non-pure Latin directed triple systems. In this paper we turn our attention to pure non-flexible and pure flexible Latin directed triple systems. ∗ Aleš Drápal supported by grant VF20102015006. † Andrew Kozlik supported by SVV-2014-260107. A. DRÁPAL ET AL. /AUSTRALAS. J. COMBIN. 62 (1) (2015), 59–75 60