  For a given inverse semigroup action on a topological space, one can
associate an \'etale groupoid. We prove that there exists a correspondence
between the certain subsemigroups and the open wide subgroupoids in case that
the action is strongly tight. Combining with the recent result of Brown et. al,
we obtain a correspondence between the certain subsemigroups of an inverse
semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.
