un 2 01 1 On connection between reducibility of an n - ary quasigroup and that of its retracts 1 Denis

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation 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. An m-ary quasigroup R 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 show that every irreducible n-ary quasigroup has an irreducible (n−1)-ary or (n−2)-ary retract; moreover, if the order is finite and prime, then it has an irreducible (n−1)-ary retract. We apply this result to show that all n-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z5 or Z7 are reducible for n ≥ 4.