Overlarge sets of resolvable idempotent quasigroups

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