@Rob: Will Theorem 4 (Identity expansion) still hold if A is of positive polarity? Or do we need to state another version if positive propositions are added to the language?

@Finn: Very useful reference, I am reading it now. I think it is what I needed. The link to Vicker's information systems strikes me, as i had previously misjudged them as not relevant to my research. Now it turns out to be intimately related.

Thanks! It seems a good direction to follow. I'll look at the material.

Exchanging $0$ and $1$ does not seem to be functorial, what would be the image of the unique morphism? It must go $F(0) \to F(1)$.