We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

We prove that for all n ≡ 13 or 37 (mod 72), there exists a biembedding of a pair of Steiner triple systems of order n in an orientable pseudosurface having precisely one regular pinch point of multiplicity 2.

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration ${{C}_{14}}$. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.

Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph $G$ is $S$-edge-colourable for a Steiner triple system $S$ if its edges can be coloured with points of $S$ in such a way that the points assigned to three edges sharing a vertex form a triple in $S$. Among others, we show that all cubic graphs are $S$-edge-colourable for every non-projective non-affine point-transitive Steiner triple system $S$.

Two Steiner triple systems (V, 𝓑) and (V, 𝓓) are orthogonal if they have no triples in common, and if for every two distinct intersecting triples {x,y,z} and {x, y, z} of 𝓑, the two triples {x,y,a} and {u, v, b} in (𝓓 satisfy a ≠ b. It is shown here that if v ≡ 1,3 (mod 6), v ≥ 7 and v ≠ 9, a pair of orthogonal Steiner triple systems of order v exist. This settles completely the question of their existence posed by O'Shaughnessy in 1968.