Continuous lattice effect algebras admitting order-continuous states

Abstract

We prove that a necessary and sufficient condition for the existence of a faithful (( o)-continuous) state on a complete modular atomic effect algebra E is the separability of E. Moreover, we generalize the famous Kaplansky theorem about order continuity of complemented complete modular lattices onto complete modular atomic effect algebras. Some other statements about the algebraic structure of modular effect algebras are shown. We prove that every chain and every block in an irreducible complete modular atomic effect algebra is finite. Moreover, every complete atomic modular effect algebra is compactly generated by finite elements.