Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4

TL;DR

This paper investigates the existence and number of transversals in reducible and order 4 quasigroups, establishing bounds and conditions for their presence in various classes of latin hypercubes.

Contribution

It provides new bounds on transversals in completely reducible quasigroups and characterizes transversal existence in quasigroups of order 4, including special cases like $ ext{Z}_4$ and $ ext{Z}_2^2$.

Findings

Most order 4 quasigroups of odd arity have transversals.

A lower bound on the number of transversals in certain classes of quasigroups.

Except for $ ext{Z}_4$, all even arity quasigroups of order 4 have transversals.

Abstract

An $n$-ary quasigroup $f$ of order $q$ is an $n$-ary operation over a set of cardinality $q$ such that the Cayley table of the operation is an $n$-dimensional latin hypercube of order $q$. A transversal in a quasigroup $f$ (or in the corresponding latin hypercube) is a collection of $q$ $(n+1)$-tuples from the Cayley table of $f$, each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011. An $n$-ary quasigroup $f$ is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary quasigroups. In this paper we investigate transversals in reducible quasigroups and in quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible quasigroups. Next we prove that, except for the iterated group $\mathbb{Z}_4$ of even arity, every $n$-ary quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in quasigroups of order 4 and odd arity and count transversals in the iterated group $\mathbb{Z}_4$ of odd arity and in the iterated group $\mathbb{Z}_2^2.$ All results of this paper can be regarded as those concerning latin hypercubes.