The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

Abstract

AbstractA ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}, is a permutation of the coordinate positions applied to the 4‐tuples of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\mathcal{B}}$\end{document}. The subgroup of conjugations under which \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} is invariant is called the conjugate invariant subgroup of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010