On the Unique Extensibility and Surjectivity of Knowledge Structures

With the S5 multi-agent epistemic logic we consider the canonical maps from Krpke structures to knowledge structures. We define a cell to be a minimal subset of knowledge structures known in common semantically by the agents. A cell has finite fanout if at every point every agent considers only a finite number of other points to be possible. We define a cell to be surjective if every Kripke structure that maps to it does so surjectively. All cells with finite fanout are surjective, but the converse does not hold. To construct a counter-example we need topological insights concerning the relationship between the logic and its semantic models. The difference between syntactic and semantic common knowledge is central to this construction.