On connection between reducibility of an nn-ary quasigroup and that of its retracts

An nn-ary operation Q:Σn→ΣQ:Σn→Σ is called an nn-ary quasigroup of order |Σ||Σ| if in the equation 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 10n−m>0 arguments.We show that every irreducible nn-ary quasigroup has an irreducible (n−1)(n−1)-ary or (n−2)(n−2)-ary retract; moreover, if the order is finite and prime, then it has an irreducible (n−1)(n−1)-ary retract. We apply this result to show that all nn-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z5Z5 or Z7Z7 are reducible for n≥4n≥4.