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...

In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups (not necessarily Abelian), an arbitrary map , and a parameter , say that is -close to a homomorphism if there is some homomorphism such that and differ on at most elements of , and say that is -far otherwise. For a given and , a homomorphism tester should ...

In this paper, we investigate the generalized Hyers-Ulam-Rassias and the Isac and Rassias-type stability of the conditional of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras. As a consequence of this, we prove the hyperstability of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras.