D. S. Krotov, V. N. Potapov, P. V. Sokolova
On reconstructing reducible n-ary quasigroups and switching subquasigroups

(1) We prove that, provided n>3, a permutably reducible n-ary quasigroup is uniquely specified by its values on the n-ples containing zero. (2) We observe that for each n,k>1 and natural r< k/2 there exists a reducible n-ary quasigroup of order k with an n-ary subquasigroup of order r. As corollaries, we have the following: (3) For each k>3 and n>2 we can construct a permutably irreducible n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>$ has double-exponential growth.