The fine structures of three idempotent Latin squares with small orders

Latin squares have wide application in communication and cipher. Denote by IdFin(v) the set of all integer pairs (t, s) for which there exist three idempotent Latin squares of order r on the same set having fine structure (t, s). We obtain that (6, 17), (4, 18)∈IdFin(7), which gives some new examples (6, 17), (4, 18)∈Fin(7). We prove that (t, 16)∉IdFin(v) with 0≤t≤3 for order v with 8 ≤ v ≤ 11, and determine the set IdFin(11).