A generalisation is presented here of the correspondence given by Greechie, between hypergraphs and finite “quantum logics”, i.e. orthomodular posets. The necessary definitions are presented clearly, then the theorem generalising Greechies result is presented. This gives conditions on a hypergraph H under which the existence of an orthomodular poset “state-isomorphic” to H can be assured.
In the last part of the paper this result is used to demonstrate some interesting examples of the “state-space properties” of some particular orthomodular posets. For example it is shown there is a “logic” whose state-space is empty, and there is a non-trivial logic whose state-space is a singleton, and there is a finite unital fully embeddable logic which is not Boolean. The author remarks that the need for such examples actually motivated the approach taken in this paper, and for those accepting the traditional tenets of “quantum logic” these will indeed prove to be interesting.