Tight inverse semigroups that are not bisimple

A semigroup is tight if each of its congruences is uniquely determined by eachof the congruence classes. Bisimple inverse semigroups are tight, and tightsemigroups are either simple or congruence-free with zero. Although congruence-freesemigroups are tight, they are not necessarily bisimple. We constructtight inverse semigroups and tight inverse monoids that are neither bisimplenor congruence-free.