Effect algebras with the maximality property

The maximality property was introduced in in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch--Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch--Piron states on an effect algebra with the maximality property is strongly order determining.