  Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic
awareness correspondence and accessibility relations modeling implicit
knowledge. They are a flexible model of unawareness, and best interpreted from
a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and
Schipper (2006, 2008) (HMS) model awareness by a lattice of state spaces and
explicit knowledge via possibility correspondences. Sublattices thereof can be
interpreted as subjective views of agents. Open questions include (1) how
implicit knowledge can be defined in HMS structures, and (2) in which way FH
structures can be extended to model the agents' subjective views. In this
paper, we address (1) by defining implicit knowledge such that it is consistent
with explicit knowledge in HMS models. We also introduce a variant of HMS
models that instead of explicit knowledge, takes implicit knowledge and
awareness as primitives. Further, we address (2) by introducing a category of
FH models that are modally equivalent relative to sublanguages and can be
interpreted as agents' subjective views depending on their awareness. These
constructions allow us to show an equivalence between HMS and FH models. As a
corollary, we obtain soundness and completeness of HMS models with respect to
the Logic of Propositional Awareness, based on a language featuring both
implicit and explicit knowledge.
