Characterization of homogeneity in orthocomplete atomic effect algebras

We prove that homogeneity can be characterized by the decompositions of sharp elements in orthocomplete atomic effect algebras. Especially, an orthocomplete atomic effect algebra is homogeneous, if and only if the non-zero coefficient of an atom in any atomic decomposition of all sharp elements is the same, if and only if the non-zero coefficient of an atom in any atomic decomposition of the unit element is the same, which confirm the name “homogeneous”. As an application, we prove the state smearing theorem for orthocomplete atomic homogeneous effect algebras.