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 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 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 } .