On reducibility of n-ary quasigroups, II

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. Q is permutably reducible ifQ(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. 1) If all the 3-ary and 4-ary retracts of an n-ary quasigroup Q are permutably reducible, then Q is permutably reducible. 2) If the n-ary quasigroup Q of finite prime order has a permutably irreducible (n− 2)-ary retract and all its (n− 1)-ary retract are permutably reducible, then Q is permutably reducible. Taking into account the result of the previous paper, we can conclude the following: every permutably irreducible n-ary quasigroup has a permutably irreducible (n − 1)-ary or (n − 2)-ary retract; every permutably irreducible n-ary quasigroup of prime order has a permutably irreducible (n− 1)-ary retract.