On lattice isomorphisms of inverse semigroups

An L-isomorphism between inverse semigroups S and T is an isomorphism between their lattices L(S) and L(T ) of inverse subsemigroups. The author and others have shown that if S is aperiodic – has no nontrivial subgroups – then any such isomorphism Φ induces a bijection φ between S and T . We first characterize the bijections that arise in this way and go on to prove that under relatively weak ‘archimedean’ hypotheses, if φ restricts to an isomorphism on the semilattice of idempotents of S, then it must be an isomorphism on S itself, thus generating a result of Goberstein. The hypothesis on the restriction to idempotents is satisfied in many applications. We go on to prove theorems similar to the above for the class of completely semisimple inverse semigroups. 2000 Mathematics Subject Classification: Primary 20M18; Secondary 08A30 Over the past quarter-century, several authors have investigated the extent to which an inverse semigroup S is determined by its lattice L(S) of inverse subsemigroups (see the survey [8] and the monograph [12]): given an L-isomorphism, that is, an isomorphism Φ : L(S) → L(T ) for some inverse semigroup T , how are S and T related? 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. Following the lead of Goberstein [4] we focus here on the situation where φE is an isomorphism (see below for a rationale for this simplification). 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. 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. In this note we first characterize the bijections so obtained, in Theorem 2.3, and then in Theorem 4.3 find a general sufficient condition in order that this bijection should be an isomorphism, improving on some results of Goberstein [4].