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