In this paper we consider $GU_{n+1}$-semigroups i.e. semigroups with the following condition: \[ (\forall x_1,x_2,\dots,x_{n+1})(\exists m)\quad(x_1x_2\dots x_{x+1})^mıangle x_1\rangle\cupangle x_2\rangle\cup\dots\cupangle x_{n+1}\rangle \] and we prove that $S$ is a $\pi$-regular $GU_{n+1}$-semigroup if and only if $S$ is a Rédei's hand of periodic nil-extensions of groups ($\pi$-groups).