| Abstract: | An n-ary operation Q:Σn→Σ is called an n-ary quasigroup of order |Σ| if in the relation x0=Q(x1,…,xn) knowledge of any n elements of x0,…,xn uniquely specifies the remaining one. 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. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible. |
