  By a "Boolean inverse semigroup" we mean an inverse semigroup whose
semilattice of idempotents is a Boolean algebra. We study representations of a
given inverse semigroup S in a Boolean inverse semigroup which are "tight" in a
certain well defined technical sense. These representations are supposed to
preserve as much as possible any trace of "Booleannes" present in the
semilattice of idempotents of S. After observing that the Vagner-Preston
representation is not tight, we exhibit a canonical tight representation for
any inverse semigroup with zero, called the "regular representation". We then
tackle the question as to whether this representation is faithful, but it turns
out that the answer is often negative. The lack of faithfulness is however
completely understood as long as we restrict to "continuous" inverse
semigroups, a class generalizing the E*-unitaries.
