Boolean closed inverse submonoid lattices (Q1375885)

This paper is a continuation of the author's article reviewed above (see Zbl 0887.20031). Let \(\breve H\) be the union of subgroups of an inverse monoid \(S\), let \(E\) be its semilattice of idempotents, and let \(H\) be the group of units. The main result is a characterization of those inverse monoids \(S\) whose lattice of closed inverse subsemigroups is boolean, \(S\) having this property if and only if \(S\) is periodic, \(E\) is isomorphic to the lattice of all co-finite sets of some set, \(H_1\) is a cyclic group whose order is a product of distinct primes, and \(\breve H\) is an \(E\)-unitary inverse subsemigroup of \(S\) with \(\breve H=EH_1\). Once again the result is interpreted in special cases including those where \(S\) is combinatorial and where \(S\) is \(E\)-unitary.