Effect algebras, witness pairs and observables

The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a witness pair for a subset of S is an object living within E that “witnesses existence” of an observable whose range includes S. We prove that there is an adjunction between the poset of all witness pairs of E and the category of all partially inverted E-valued observables.