In this paper, the notion of $n$-anti-inverse semigroups is introduced (for binary anti-in verse semigroups see [1]). $A_n$ $n$-ary semigroup $(S,(\,))$ is an $n$-anti-inverse semigroup iff the following condition is satisfied \[ (\forall aı \underset{n}S)(\exists bı S)(verset{n-1}{a}bverset{n-1}{a}=b\wedge bverset{2n-3}{a}b=a) \] where $x^n_1$ is $(x_1,x_2,\dots,x_n)$ and $x$ is $x_1=x_2=\dots=x_n=x$. Theorem 2.1. gives a characterization of the above semigroups. It is also proved that a semigroup $S$ is the union of its right ideal $S_a$, where $S_a=\{x\in S\mid\overset{4n-4}ax=x\}$. For all right ideal $S_a$ there is a binary operation $\circ$ such that $(S_a,\circ)$ is a groupoid having a left unit such that $(x^n_1)=\overset{n}a\circ(\ldots((x_1\circ x_2)\circ x_3)\dots)\circ x_n)$ oindent (Theorem 2.2.). If $S$ is a commutative $n$-anti-inverse semigroup, then groupoids $(S,\circ)$, for $a\in S$, are semigroups with a unit. (Theorem 2.3.).