  We explore the relation between lattice versions of strict singularity for
operators from a Banach lattice to a Banach space. In particular, we study when
the class of disjointly strictly singular operators, those not invertible on
the span of any disjoint sequence, coincides with that of lattice strictly
singular operators, i.e. those not invertible on any (infinite dimensional)
sublattice. New results are given which help to clarify the existing relation
between these two classes.
