On irreducible n-ary quasigroups with reducible retracts

Abstract

An n-ary operation q:A^n->A is called an n-ary quasigroup of order |A| 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_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)}) where p and r are (n-k+1)-ary and k-ary quasigroups, s 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 a permutably irreducible n-ary quasigroup such that all its (n-1)-ary retracts are permutably reducible is an open question; however, there are nonexistence results for 5-ary and 7-ary quasigroups of order 4. Keywords:n-ary quasigroups, n-quasigroups, reducibility, Seidel switching, two-graphs