On irreducible n-ary quasigroups with reducible retracts

An n-ary operation Q : Sigma(n) -> Sigma is called an n-ary quasigroup of order vertical bar Sigma vertical bar if in x0 = Q(x(1), ... , x(n)) knowledge of any n elements of x(0), ... , x(n) uniquely specifies the remaining one. An n-ary quasigroup Q is permutably reducible if Q(x(1), ... , x(n)) = P(R(x(sigma(1)), ... , x(sigma(k))), x(sigma(k+1)), ... , x(sigma(n))) where P and R are (n - k + 1)-ary and k-ary quasigroups, sigma is a permutation, and 1 < k < n. For even n we construct a permutably irreducible n-ary quasigroup of order 4r such that all its retracts obtained by fixing one variable are permutably reducible. We use a partial Boolean function that satisfies similar properties. For odd n the existence of permutably irreducible n-ary quasigroups with permutably reducible (n - I)-ary retracts is an open question; however, there are nonexistence results for 5-ary and 7-ary quasigroups of order 4. (c) 2007 Elsevier Ltd. All rights reserved.