  The standard semantics of multi-agent epistemic logic S5 is based on Kripke
models whose accessibility relations are reflexive, symmetric and transitive.
This one dimensional structure contains implicit higher-dimensional information
beyond pairwise interactions, that we formalized as pure simplicial models in a
previous work (Information and Computation, 2021). Here we extend the theory to
encompass simplicial models that are not necessarily pure. The corresponding
class of Kripke models are those where the accessibility relation is symmetric
and transitive, but might not be reflexive. Such models correspond to the
epistemic logic KB4 . Impure simplicial models arise in situations where two
possible worlds may not have the same set of agents. We illustrate it with
distributed computing examples of synchronous systems where processes may
crash.
