Overlarge sets of resolvable idempotent quasigroups

An idempotent quasigroup (X, ◦) of order v is called resolvable (denoted by RIQ(v)) if the set of v(v − 1) non-idempotent 3-vectors {(a, b, a ◦ b) : a, b ∈ X, a ̸= b} can be partitioned into v−1 disjoint transversals. An overlarge set of idempotent quasigroups of order v, briefly by OLIQ(v), is a collection of v + 1 IQ(v)s, with all the nonidempotent 3-vectors partitioning all those on a (v + 1)-set. An OLRIQ(v) is an OLIQ(v) with each member IQ(v) being resolvable. In this paper, it is established that there exists an OLRIQ(v) for any positive integer v ≥ 3, except for v = 6, and except possibly for v ∈ {10, 11, 14, 18, 19, 23, 26, 30, 51}. An OLIQ(v) is another type of restricted OLIQ(v) in which each member IQ(v) has an idempotent orthogonal mate. It is shown that an OLIQ (v) exists for any positive integer v ≥ 4, except for v = 6, and except possibly for v ∈ {14, 15, 19, 23, 26, 27, 30}.