A Steiner triple system, a \(STS\), (a Mendelsohn triple system, a \(MTS\), a directed triple system, a \(DTS\)) is a pair \((V,\mathcal{B})\) where \(B\) is a set of points and \(\mathcal{B}\) is a collection of triples (cyclically oriented triples, transitively oriented triples) of distinct points from \(V\) so that each pair (ordered pair) of points from \(V\) occurs in exactly one triple in \(\mathcal{B}\). Define an operation \(\cdot \) on \(V\) by \(x\cdot x=x\) and \(x\cdot y=z\) if \(z\) is the third point in the triple in \( \mathcal{B}\) containing the pair \(\{x,y\}\) (ordered pair \((x,y)\)). It is well-known that \((V,\cdot )\) is a quasigroup if \((V,\mathcal{B})\) is a \(STS\) system or a \(MTS\), but this does not have to be true in general for a \( DTS\). The authors initiate the study of directed triple systems for which \( (V,\cdot )\) forms a quasigroup.