On Lattice Isomorphisms of Inverse Semigroups, II

A lattice isomorphism between inverse semigroups S and T is an isomorphism between their lattices of inverse subsemigroups. When S is combinatorial, it has long been known that a bijection is induced between S and T. Various authors have introduced successively weaker "archimedean" hypotheses under which this bijection is necessarily an isomorphism, naturally inducing the original lattice isomorphism. Since lattice-isomorphic groups need not have the same cardinality, extending these techniques to the non-combinatorial case requires some means of tying the subgroups to the rest of the semigroup. Ershova showed that if S has no nontrivial isolated subgroups (subgroups that form an entire D-class) then again a bijection exists between S and T. Recently, this technique has been successfully exploited, by Goberstein for fundamental inverse semigroups and by the author for completely semisimple inverse semigroups, under two different finiteness hypotheses. In this paper, we derive further properties of Ershova's bijection(s) and formulate a "quasi-connected" hypothesis that enables us to derive both Goberstein's and the author's earlier results as corollaries.