Abstract Isotopies and autotopies of n -ary groups are described. As a consequence, we obtain various characterizations the group automorphisms groups. We also determine number given group.

We devise a condition strictly between the existence of an $n$-ary and $n{+}1$-ary near-unanimity term. evaluate exactly distributivity modularity levels implied by such condition.

Hyperstructure theory was born in 1934 when Marty [19] defined hypergroups as a generalization of groups. Let H be a non-empty set and let ℘∗(H) be the set of all non-empty subsets of H. A hyperoperation on H is a map ◦ : H ×H −→ ℘∗(H) and the couple (H, ◦) is called a hypergroupoid. If A and B are non-empty subsets of H, then we denote A◦B = ∪ a∈A, b∈B a◦b, x◦A = {x}◦A and A◦x = A◦{x}. Under c...

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-a...

An n-ary operation q : Σn → Σ is called an n-ary quasigroup of order |Σ| if in 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. ...