The existence of overlarge sets of idempotent quasigroups

A idempotent quasigroup (Q, o) of order n is equivalent to an n(n - 1) x 3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n + 1)-set. Denote by T(n + 1) the set of (n + 1)n(n - 1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n + 1) into n + 1 n(n - 1) x 3 partial orthogonal arrays A(x), x is an element of X based on X \ {x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.