Extensions of ordering sets of states from effect algebras onto their MacNeille completions

The notion of an effect algebra was presented by Foulis and Bennett in (Foulis, Bennett, [3] 1994). The definition was motivated by giving an algebraic description of a logic of quantum effects E(H), i.e. the set of all positive self-adjoint operators between zero and identity operator I in a separable complex Hilbert space H. On E(H) was defined a partial operation A ⊕ B = A + B iff A + B ≤ I with meaning of an orthogonal disjunction. Quantum effects in studies of quantum mechanics correspond to yes-no measurements that may be unsharp. An equivalent structure called D-poset has been introduced by Kôpka and Chovanec ([7] 1992, [6] 1994).