ON LATTICE ISOMORPHISMS OF INVERSE SEMIGROUPS

Abstract

An ${\cal L}$-isomorphism between inverse semigroups $S$ and $T$ is an isomorphism between their lattices ${\cal L}(S)$ and ${\cal L}(T)$ of inverse subsemigroups. The author and others have shown that if $S$ is aperiodic – has no nontrivial subgroups – then any such isomorphism $\Phi$ induces a bijection $\phi$ 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 $\phi$ 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.