  The fundamental problem in hierarchical supervisory control under partial
observation is to find conditions preserving observability between the original
(low-level) and the abstracted (high-level) plants. Two conditions for
observable specifications were identified in the literature -- observation
consistency (OC) and local observation consistency (LOC). However, the
decidability of OC and LOC were left open. We show that both OC and LOC are
decidable for regular systems. We further show that these conditions do not
guarantee that supremal (normal or relatively observable) sublanguages computed
on the low level and on the high level always coincide. To solve the issue, we
suggest a new condition -- modified observation consistency -- and show that
under this condition, the supremal normal sublanguages are preserved between
the levels, while the supremal relatively observable high-level sublanguage is
at least as good as the supremal relatively observable low-level sublanguage,
i.e., the high-level solution may be even better than the low-level solution.
