  The satisfiability problem of hybrid logics with the downarrow binder is
known to be undecidable. This initiated a research program on decidable and
tractable fragments. In this paper, we investigate the effect of restricting
the propositional part of the language on decidability and on the complexity of
the satisfiability problem over arbitrary, transitive, total frames, and frames
based on equivalence relations. We also consider different sets of modal and
hybrid operators. We trace the border of decidability and give the precise
complexity of most fragments, in particular for all fragments including
negation. For the monotone fragments, we are able to distinguish the easy from
the hard cases, depending on the allowed set of operators.
