The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the ag manifolds of the group. For this purpose a convexity theory, called here B-convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a B-convex set.3531251655184