On reducibility of n-ary quasigroups

An n  -ary operation Q:Σn→ΣQ:Σn→Σ is called an n  -ary quasigroup of order |Σ||Σ| if in the relation x0=Q(x1,…,xn)x0=Q(x1,…,xn) knowledge of any n   elements of x0,…,xnx0,…,xn uniquely specifies the remaining one. Q   is permutably reducible if Q(x1,…,xn)=P(R(xσ(1),…,xσ(k)),xσ(k+1),…,xσ(n))Q(x1,…,xn)=P(R(xσ(1),…,xσ(k)),xσ(k+1),…,xσ(n)) where P and R   are (n-k+1)(n-k+1)-ary and k  -ary quasigroups, σσ is a permutation, and 10n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q   belongs to {3,…,n-3}{3,…,n-3}, then Q is permutably reducible.