Some classes of extended Mendelsohn triple systems and numbers of common blocks

An extended Mendelsohn triple system of the order v with a idempotent element (EMTS(v, a)) is a collection of cyclically ordered triples of the type [x, y, z], [x, x, y] or [x, x, x] chosen from a v-set, such that every ordered pair (not necessarily distinct) belongs to only one triple and there are a triples of the type {x, x, x}. If such a design with parameters v and a exist, then they will have b(v,a) blocks, where b(v,a) = (v(2) + 2a)/3. A necessary and sufficient condition for the existence of EMTS(v, 0) and EMTS(v, 1) are v equivalent to 0 (mod 3) and v not equivalent to 0 (mod 3), respectively. In this paper, we have constructed two EMTS(v,0)'s such that the number of common triples is in the set {0, 1, 2, ..., b(v,0) - 3, b(v,0)}, for v equivalent to 0 (mod 3). Secondly, we have constructed two EMTS(V,l)'s such that the number of common triples is in the set {0, 1, 2,...,b(v,1) - 2, b(v,1)}, for v not equivalent to 0 (mod 3).