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=xx⋅x=x and x⋅y=zx⋅y=z, where zz is the third point in the block containing the pair {x,y}{x,y}. The same is true for a Mendelsohn triple system, where the pair (x,y)(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. We call these Latin directed triple systems, and in this paper we begin the study of their existence and properties.