Smearings of States Defined on Sharp Elements Onto Effect Algebras

Abstract

In some sense, a lattice effect algebra E is a smeared orthomodular lattice S(E), which then becomes the set of all sharp elements of the effect algebra E. We show that if E is complete, atomic, and (o)-continuous, then a state on E exists iff there exists a state on S(E). Further, it is shown that such an effect algebra E is an algebraic lattice compactly generated by finite elements of E. Moreover, every element of E has a unique basic decomposition into a sum of a sharp element and a ⊕-orthogonal set of unsharp multiples of atoms.