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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...
The main result of this paper is Theorem 3.3: Every concrete logic (i.e., every set-representable orthomodular poset) can be enlarged to a concrete logic with a given automorphism group and with a given center. Since every sublogic of a concrete logic is concrete, too, and since not every state space of a (general) quantum logic is affinely homeomorphic to the state space of a concrete logic [8], our result seems in a sense the best possible. Further, we show that every group is an automorphism...
We introduce the notion of p-ideal of a QMV-algebra and we prove that the class of all p-ideals of a QMV-algebra M is in one-to-one correspondence with the class of all congruence relations of M.
An equivalent definition of compatibility in pseudo-effect algebras is given, and its relationships with central elements are investigated. Furthermore, pseudo-MV-algebras are characterized among pseudo-effect algebras by means of compatibility.
We present three results stating when a concrete (=set-representable) quantum logic with covering properties (generalization of compatibility) has to be a Boolean algebra. These results complete and generalize some previous results [3, 5] and answer partiallz a question posed in [2].
Effect basic algebras (which correspond to lattice ordered effect algebras) are studied. Their ideals are characterized (in the language of basic algebras) and one-to-one correspondence between ideals and congruences is shown. Conditions under which the quotients are OMLs or MV-algebras are found.
Given a real separable Hilbert space H, we denote with S = {E(n) | n belongs to N} a sequence of closed linear subspaces of H.In previous papers, the strong, weak, a--> and b--> convergences are defined and characterized. Now, given a sequence S with strong, weak, a--> or b--> limit, and a linear operator of H, A, the sequence AS is studied.
A degree of probabilistic dependence is introduced in the classical logic using the Frank family of -norms known from fuzzy logics. In the quantum logic a degree of quantum dependence is added corresponding to the level of noncompatibility. Further, in the case of the fuzzy logic with -states, (resp. -states) the consideration turned out to be fully analogous to (resp. considerably different from) the classical situation.
Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital...
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