Inverse Sets and Inverse Correspondences Over Inverse Semigroups

In this paper, we introduce notions called inverse set and inverse correspondence over inverse semigroups. These are analogies of Hilbert $C^{\ast }$-modules and \Ccorrs in the $C^{\ast }$-algebra theory. We show that inverse semigroups and inverse correspondences form a bicategory. In this bicategory, two inverse semigroups are equivalent if and only if they are Morita equivalent.