Self-orthogonal mendelsohn triple systems

A Mendelsohn triple system of order v, briefly MTS(v), is a pair (X, B) where X is a v-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c), and (c, a). An MTS(v) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order v. An MTS(v) is called self-orthogonal, denoted briefly by SOMTS(v), if its associated semisymmetric Latin square is self-orthogonal. It is well known that an MTS(v) exists if and only if v ≡ 0 or 1 (mod 3) except v ≠ 6. It is also known that a SOMTS(v) exists for all v ≡ 1 (mod 3) except v = 10 and that a SOMTS(v) does not exist for v = 3, 6, 9, and 12. In this paper it is shown that a SOMTS(v) exists for all v ⩾ 15, where v ≡ 0 (mod 3), except possibly for v = 18.