Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an “effect” and show that observables are related to effect-valued measures. Finally, (...) we define “operational” transformations from M(H) to itself and show that they can be described as a certain combination of effects. (shrink)

Croc lattices are of central importance in the mathematical foundation of quantum mechanics. They reflect the structure of the logical calculus of propositions concerning microphenomena. In this context, the mathematical consequences resulting from the special structure of the lattice are of particular interest. Beginning with the notion of hemimorphism of a croc lattice, we get, in a natural way, an extension of the structure of the lattice by the Baer *-semigroup. We embed the Baer *-semigroup in an algebra with generalized (...) *- and ′-operations and prove some theorems as consequences of the particular properties of the underlying croc lattice. (shrink)