On irreducible -ary quasigroups with reducible retracts

An n -ary operation Q : Σ n → Σ is called an n -ary quasigroup of order | Σ | if in x 0 = 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 σ ( 1 ) , … , x σ ( k ) ) , x σ ( k + 1 ) , … , x σ ( n ) ) where P and R are ( n − k + 1 ) -ary and k -ary quasigroups, σ is a permutation, and 1 < k < n . For even n we construct a permutably irreducible n -ary quasigroup of order 4 r 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 − 1 ) -ary retracts is an open question; however, there are nonexistence results for 5-ary and 7-ary quasigroups of order 4.