Algebra Universalis. DOI 10.1007/s00012-009-0013-3 The original publication is available at www.springerlink.com. Effect algebras with the maximality property

The maximality property was introduced in [9] in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete ortho-modular 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. 1 Basic notions Effect algebras as generalizations of orthomodular posets (quantum logics) are stud-ied in the axiomatics of quantum systems—see, e.g., [1, 2]. 1.1.Definition. An effect algebra is an algebraic structure (E,⊕,0,1) such that E is a set, 0 and 1 are distinct elements of E and ⊕ is a partial binary operation on E such that for every a, b, c ∈ E the following conditions hold