For an n-ary groupoid we nd the neccesity and sucient conditions under which all its transformations are endomorphisms. It is known [1] that a semigroup in which each transformation is an endomorphism, is a left or right zero semigroup. Below we generalize this result to the case of n-ary groupoids. Let (G, o) be an n-ary groupoid, i.e., a nonempty set G with an n-ary operation o. Such groupoid is also called an n-groupoid (cf. [2]). An element 0 2 G is called a k-zero, where k 2 {1, 2, . . . , n}, of an n-groupoid (G, o), if o(x1, . . . , xk−1, 0, xk 1, . . . , xn) = 0 holds for all x1, . . . , xk−1, xk 1, . . . , xn 2 G. An n-groupoid in which each element is a k-zero is called an n-groupoid of k-zeros or a k-zero n-groupoid. Following [3], an n-groupoid (G, o) in which o(x1, . . . , xn) 2 {x1, . . . , xn} for any x1, . . . , xn 2 G is called quasitrivial.