Representations of Inverse Semigroups and Pre-grading C *-Algebra

In this work, we present a definition of an inverse semigroup, Pr(S), which is associated to an inverse semigroup S. Also, we show the existence of a kind of correspondence between partial representations of S and representations of Pr(S) on a Hilbert space. Some results of graded C *-algebras over a group are extended to pre-grading C *-algebras over inverse semigroups.