An -ary quasigroup of order is an -ary operation over a set of cardinality such that the Cayley table of the operation is an -dimensional latin hypercube of order . A transversal in a quasigroup (or in the corresponding latin hypercube) is a collection of
-tuples from the Cayley table of , each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011.
An -ary quasigroup 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 of even arity, every -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 of odd arity and in the iterated group
All results of this paper can be regarded as those concerning latin hypercubes.