Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3 ∣ v and v ⁄ ≡ 2 ( mod 4 ) . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems LRMTS ( v ) s, which improves an earlier version of tripling construction by Kang. As an application we obtain an LRMTS ( 4 ⋅ 3 n ) for any integer n ≥ 1 , which provides an infinite family of even orders.