Hyper-commutators in effect algebras

The hyper-commutators in effect algebras as a generalization of commutators in lattice effect algebras are defined, and their properties are studied. We prove that a nonzero element is in some hyper-commutator if and only if it is a torsion element in effect algebras with the maximality property. Thus, homogeneity can be characterized by hyper-commutators. Some properties of Riesz ideals related to hyper-commutators are also studied. As an application, we prove that commutators are sharp in lattice effect algebras and every block-finite lattice effect algebra is commutator-finite.