Zero-semidistributive inverse semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice LF (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a, b is an element of S/sigma with ab not equal 1, there exist nonzero integers n and m such that (ab)(m) = a(n) or (ab)(m) = b(n), where sigma is the minimum group congruence on S.