The construction of large sets of idempotent quasigroups (Q1102378)

The maximum number of idempotent quasigroups of order n which pairwise agree on the main diagonal only is n-2, is called a large set of idempotent quasigroups of order n. In this paper is introduced the construction of a large set of idempotent quasigroups of order n, but not for every n. So, the main result is Theorem: There exists a large set of idempotent orthogonal arrays of order n for every \(n\geq 3\) except \(n=6\) (for which no such collection exists) and possibly \(n=14\) and 62. Additionally, the known spectrum for large sets of Mendelsohn quasigroups is improved.