Overlarge sets of resolvable idempotent quasigroups

An idempotent quasigroup (X, o) of order u is called resolvable (denoted by RIQ(nu)) if the set of nu(nu - 1) non-idempotent 3-vectors {(a, b, a o b) : a, b is an element of X, a not equal b} can be partitioned into nu - 1 disjoint transversals. An overlarge set of idempotent quasigroups of order nu, briefly by OLIQ(v), is a collection of nu + 1 IQ(nu)s, with all the nonidempotent 3-vectors partitioning all those on a (nu - 1)-set. An OLRIQ(nu) is an OLIQ(nu) with each member IQ( nu) being resolvable. In this paper, it is established that there exists an OLRIQ(nu) for any positive integer nu >= 3, except for nu = 6, and except possibly for nu is an element of {10, 11, 14, 18, 19. 23, 26, 30. 51}. An OLIQ degrees (nu) is another type of restricted OLIQ(nu) in which each member IQ(nu) has an idempotent orthogonal mate. It is shown that an OUT( v) exists for any positive integer v >= 4, except for nu = 6, and except possibly for nu is an element of {14, 15, 19, 23, 26, 27, 30). (C) 2012 Elsevier B.V. All rights reserved.