Abstract:

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra $P$ are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra $P$ is a union of generalized MV-effect algebras and every generalized homogeneous effect algebra is a union of its maximal sub-generalized effect algebras with hereditary Riesz decomposition property (blocks). Properties of sharp elements, the center and center of compatibility of $P$ are shown. We prove that on every generalized MV-effect algebra there is a bounded orthogonally additive measure.