  (1) We prove that, provided n>=4, 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>=2 and 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>=4 and n>=3 we can construct a permutably irreducible
n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>3
has double-exponential growth as n tends to infinity; it is greater than exp
exp(n ln[k/3]) if k>=6, and exp exp(n (ln 3)/3 - 0.44) if k=5.

Combinatorics;Group Theory