We prove that the interval topology of an Archimedean atomic lattice effect algebra E is Hausdorff whenever the set of all atoms of E is almost orthogonal. In such a case E is order continuous. If moreover E is complete then order convergence of nets of elements of E is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on E corresponding to compact and cocompact elements. For block-finite Archimedean atomic lattice effect algebras the equivalence of almost orthogonality and s-compact generation is shown. As the main application we obtain a state smearing theorem for these effect algebras, as well as the continuity of ⊕-operation in the order and interval topologies on them.
We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element E of which has atomic center C(E) or the subset S(E) of all sharp elements, resp. the center of compatibility B(E) or every block M of E. The atomicity of E or its sub-lattice effect algebras C(E), S(E), B(E) and blocks M of E is very useful equipment for the investigations of its algebraic and topological properties, the existence or smearing of states on E, questions about isomorphisms and so. Namely we touch the families of complete lattice effect algebras, or lattice effect algebras with finitely many blocks, or complete atomic lattice effect algebra E with Hausdorff interval topology.
The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (Rl-monoids) are common generalizations of BL-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded Rl-monoids.
An effect algebraic partial binary operation ⊕ defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion Eˆ of E there exists an effect algebraic partial binary operation ⊕ˆ then ⊕ˆ need not be an extension of ⊕. Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ˆ existing on Eˆ is an extension of ⊕ defined on E. Further we show that such ⊕ˆ extending ⊕ exists at most one.
Effect algebras are very natural logical structures as carriers of probabilities and states. They were introduced for modeling of sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Nevertheless, there are effect algebras without any state, and questions about the existence (for non-modular) are still unanswered. We show that every Archimedean atomic lattice effect algebra with at most five blocks (maximal MV-subalgebras) has at least one state, which can be obtained by "State Smearing Theorem'' from a state on its sharp elements.