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 aperiodic, 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-aperiodic 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 ‘archimedean’ 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. 2000 Mathematics Subject Classification: Primary 20M18; Secondary 08A30 This paper is a sequel to the author’s paper [11] and, at the same time, an extension of a recent paper by Goberstein [5], on the extent to which an inverse semigroup S is determined by its lattice L(S) of inverse subsemigroups: given an L-isomorphism, that is, an isomorphism Φ : L(S) → L(T ) for some inverse semigroup T , how are S and T related? For surveys on this topic, see [10] and [14]. It is easily seen that since Φ restricts to an L-isomorphism between their respective semilattices of idempotents, ES and ET , it induces a bijection φE between them. Continuing the context of the cited papers, we focus on the situation where φE is an isomorphism, which occurs in a surprising number of contexts, for example (see [8]) when S is simple, when S is E-unitary with no trivial J -classes or, except in a singular case, when Φ is induced by an isomorphism between the partial automorphism semigroups of S and T . It has long been known that φE extends to a bijection φ : ES ∪NS → ET ∪NT , where NS denotes the set of elements that belong to no subgroup of S. This is termed the “partial base bijection” in [14]. In the aperiodic (or “combinatorial”) case where, by definition, all subgroups are trivial, φ is then a bijection between S and T . In turn, φ induces Φ in the obvious way. It was shown by Ershova (see [14]) that as long as S has no nontrivial isolated subgroups, then the partial bijection φ can be extended to a “base bijection”, again denoted φ, between S and T . A priori, φ may extend non-uniquely. In this paper, we study the properties of the partial base bijection (in §1) and of base bijections (§2) under a finiteness hypothesis termed quasi-connectedness, to be defined shortly. We thereby show that under this hypothesis (together with the hypothesis on φE and that on subgroups), if S either (a) is fundamental or (b) is completely semisimple with at least three idempotents in each D-class, any partial base bijection, as above, is necessarily an isomorphism. We then prove that the “quasi-archimedean” condition considered by the author in [11] and the “tightly connected” condition considered by Goberstein in [5] each implies quasi-connectedness, thereby deducing a main result of each of the cited papers.