In Part I of this paper we showed that the study of Co-isomorphisms of inverse semigroups, that is, isomorphisms between the lattices of convex inverse subsemigroups of two such semigroups, can be reduced to consideration of those that induce an isomorphism between the respective semilattices of idempotents. We go on here to prove two somewhat complementary theorems. We show that the inverse semigroups Co-isomorphic to the bicyclic semigroup are precisely the well-known simple, aperiodic, inverse ω-semigroups B_d. And we show that for completely semisimple inverse semigroups (those that contain no bicyclic subsemigroup), Co-isomorphisms that induce an isomorphism between the semilattices of idempotents of the respective inverse semigroups are entirely equivalent to L-isomorphisms with the same property. (An L-isomorphism is an isomorphism between the lattices of all inverse subsemigroups.) Combining this result, known results on L-isomorphisms and the main theorem of Part I yields a complete determination of Co-isomorphisms for broad classes of semigroups. For some slightly narrower classes it is known that every Co-isomorphism of necessity induces an isomorphism on the semilattice of idempotents, yielding theorems on their Co-determinability.
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