  In this paper we define and study dimension generalized effect algebras
(DGEAs), i.e., Dedekind orthocomplete and centrally orthocomplete generalized
effect algebras equipped with a dimension equivalence relation. Our theory is a
bona fide generalization of the theory of dimension effect algebras (DEAs),
i.e., it is formulated so that, if a DGEA happens to be an effect algebra
(i.e., it has a unit element), then it is a DEA. We prove that a DGEA
decomposes into type I, II, and III DGEAS in a manner analogous to the type
I/II/III decomposition of a DEA.
