States, Uniformities and Metrics on Lattice Effect Algebras

Effect algebras [6] (or, equivalent in some sense, D-posets [13], [14]) were introduced as carriers of states or probability measures in the quantum (or fuzzy) probability theory (see [10], [11], [13]). Thus elements of these structures represent quantum effects or fuzzy events which have yes-no character that may be unsharp or imprecise. Unfortunately, there are even finite effect algebras admitting no states hence also no probabilities [19]. Moreover, a state on an effect algebra need not be subadditive. It was proved in [20] that a state on a lattice effect algebra is subadditive iff it is a valuation. Further, if a faithful (i.e., non-zero at non-zero elements) valuation on an effect algebra E exists then E is modular and separable [20]. Conversely, on every complete modular atomic effect algebra there exists an (o)-continuous state [18], [21]. The aim of this paper is to bring some topological properties of lattice (or complete) effect algebras on which states, order-continuous states or valuations exist. Namely, we study properties of order and interval topologies of such effect algebras. Further we show relations of these topologies to uniform or metric topologies induced by states or valuations on them.