Basic decomposition of elements and Jauch-Piron effect algebras

A logical structure of propositions associated with a physical system,which can be xperimentally tested,form in the clasical case a Boolean algebra and in a quantum mechanical case an orthomodular lattice or poset (a quantum logic). Recently new logical structures for the presence of propositions, properties,questions or events with fuzziness,uncertainity or unsharpness has been introduced: Effect algebras, introduced by Foulis and Bennett (1994) for studing quantum effects ;and D-posets, introduced by Kopka (1992) and Kopka and Chovanec (1994) for studing fuzzy events. Lattice ordered effect algebras include Boolean algebras and orthomodular lattices as well as MV-algebras employed by Chang (1958) in the analysis of many-valued logics. In spite of the fact that carriers of probabilities in quantum or fuzzy probability theory are effect algebras, there are even finite effect algebras admitting no states and hence no probabilities (Greechie,Riecanova). We show some familes of effect algebras admitting order-continuous or sigma-additive states. We show that every element of a complete atomic effect algebra E has a unic basic decomposition into a sum of a sharp element and nonsharp multiples of isotropic atoms of E. Consequently we obtain for such effect algebras "The Smearing Theorem for States ",establishes that every order-continuous state existing on sharp elements of E can be extended to a state on E. The Jauch-Piron condition for states appears at many places in the axiomatics of quantum systems. For a sigma-complete separable atomic effect algebra E we prove that E is unital and Jauch-Piron effect algebra if and only if the set S(E) of all sharp elements of E is a unital Jauch-Piron orthomodular lattice and for finite E , the S(E) is a Boolean algebra.