On irreducible nn-ary quasigroups with reducible retracts

An nn-ary operation Q:Σn→ΣQ:Σn→Σ is called an nn-ary quasigroup of order |Σ||Σ| if in x0=Q(x1,…,xn)x0=Q(x1,…,xn) knowledge of any nn elements of x0,…,xnx0,…,xn uniquely specifies the remaining one. An nn-ary quasigroup QQ is permutably reducible if Q(x1,…,xn)=P(R(xσ(1),…,xσ(k)),xσ(k+1),…,xσ(n))Q(x1,…,xn)=P(R(xσ(1),…,xσ(k)),xσ(k+1),…,xσ(n)) where PP and RR are (n−k+1)(n−k+1)-ary and kk-ary quasigroups, σσ is a permutation, and 1