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,[email protected]?b):a,[email protected]?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 non-idempotent 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 [email protected]?{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 [email protected]?{14,15,19,23,26,27,30}.