On irreducible n-ary quasigroups with reducible retracts

An n-ary operation q : Σn → Σ is called an n-ary quasigroup of order |Σ| if in x0 = q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. An n-ary quasigroup 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 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 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.