Observables on σ-Frame effect Algebras as Upper Semicontinuous Functions

We showed that the set of observables O(E) on a σ-frame effect algebra E is in one-to-one correspondence with the set USC(is) of point-free versions of real upper semicontinous functions on the order reduct of E. The Olson order and the sum on O(E) are represented by the usual order and the sum of real upper semicontinuous functions. Sharp observables SO(E) are then described by continuous functions on E and this correspondence is one-to-one in the case when E forms an MV-algebra.