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In analogy with effect algebras, we introduce the test spaces and -test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between -algebras and -test spaces.
If element of a lattice effect algebra is central, then the interval is a lattice effect algebra with the new top element and with inherited partial binary operation . It is a known fact that if the set of central elements of is an atomic Boolean algebra and the supremum of all atoms of in equals to the top element of , then is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion of which is its extension...
Lattice effect algebras generalize orthomodular lattices and -algebras. We describe all complete modular atomic effect algebras. This allows us to prove the existence of ordercontinuous subadditive states (probabilities) on them. For the separable noncomplete ones we show that the existence of a faithful probability is equivalent to the condition that their MacNeille complete modular effect algebra.
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