Let $S={a,b,c,dots }$ and $ Gamma = {alpha,eta,gamma,dots }$ be two nonempty sets. $S$ is called a $ Gamma $-semigroup if $ a alpha b in S $, for all $alpha in Gamma $ and $a,b in S $ and $(a alpha b)eta c =a alpha (beta c)$, for all $a,b,c in S $ and for all $ alpha,eta in Gamma $. An element $ein S$ is said to be an $alpha $-idempotent for some $alpha in Gamma $ if $ealpha e = e$. A $Gamma $-semigroup $S$ is called an $E$-inversive $Gamma $-semigroup if for each $ain S$ there exist $xin S$ and $alpha in Gamma $ such that $aalpha x$ is a $eta $-idempotent for some $eta in Gamma $. A $Gamma $-semigroup is called a right $E$- $Gamma $-semigroup if for each $alpha $-idempotent $e$ and $eta $-idempotent $f$, $ealpha f$ is a $eta $-idempotent. In this paper we investigate different properties of $E$-inversive $Gamma $-semigroup and right $E$-$Gamma $-semigroup.

Keywords: E-inversive $Gamma $-semigroup, Right E-$Gamma $-semigroup, semidirect product