Every inverse semigroup possesses a natural partial order and therefore convexity with respect to this order is of interest. We study the extent to which an inverse semigroup is determined by its lattice of convex inverse subsemigroups; that is, if the lattices of two inverse semigroups are isomorphic, how are the semigroups related? We solve this problem completely for semilattices and for inverse semigroups in general reduce it to the case where the lattice isomorphism induces an isomorphism between the semilattices of idempotents of the semigroups. For many inverse semigroups, such as the monogenic ones, this case is the only one that can occur. In Part II, a study of the reduced case enables us to prove that many inverse semigroups, such as the free ones, are strictly determined by their lattices of convex inverse subsemigroups, and to show that the answer obtained here for semilattices can be extended to a broad class of inverse semigroups, including all finite, aperiodic ones.
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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