On reducibility of n-ary quasigroups

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , 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) ) where P and R are (n− k+1)-ary and kary quasigroups, σ is a permutation, and 1 < k < n. Anm-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n−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}, then Q is permutably reducible.