Uniqueness of bounded observables
State space properties of finite logics
Announcements of new results
Quantum logics representable as kernels of measures
Two-valued states on a concrete logic and the additivity problem
Descriptions of state spaces of orthomodular lattices (the hypergraph approach)
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...
Integration on generalized measure spaces
On the integration on -classes
Program for generating fuzzy logical operations and its use in mathematical proofs
Fuzzy logic is one of the tools for management of uncertainty; it works with more than two values, usually with a continuous scale, the real interval . Implementation restrictions in applications force us to use in fact a finite scale (finite chain) of truth degrees. In this paper, we study logical operations on finite chains, in particular conjunctions. We describe a computer program generating all finitely-valued fuzzy conjunctions (-norms). It allows also to select these -norms according to...
Considering uncertainty and dependence in Boolean, quantum and fuzzy logics
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.
Validation sets in fuzzy logics
The validation set of a formula in a fuzzy logic is the set of all truth values which this formula may achieve. We summarize characterizations of validation sets of -fuzzy logics and extend them to the case of -fuzzy logics.
Automorphisms of concrete logics
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...
Two-valued measures on -classes
The σ-complete MV-algebras which have enough states
We characterize Łukasiewicz tribes, i.e., collections of fuzzy sets that are closed under the standard fuzzy complementation and the Łukasiewicz t-norm with countably many arguments. As a tool, we introduce σ-McNaughton functions as the closure of McNaughton functions under countable MV-algebraic operations. We give a measure-theoretical characterization of σ-complete MV-algebras which are isomorphic to Łukasiewicz tribes.
A characterization of tribes with respect to the Łukasiewicz -norm
We give a complete characterization of tribes with respect to the Łukasiewicz -norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz -norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental -norms, e. g., for the product -norm.
On the permanence properties of interval homogeneous orthomodular lattices
A Cantor-Bernstein theorem for -complete MV-algebras
The Cantor-Bernstein theorem was extended to -complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to -complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.
On interval homogeneous orthomodular lattices
An orthomodular lattice is said to be interval homogeneous (resp. centrally interval homogeneous) if it is -complete and satisfies the following property: Whenever is isomorphic to an interval, , in then is isomorphic to each interval with and (resp. the same condition as above only under the assumption that all elements , , , are central in ). Let us denote by Inthom (resp. Inthom) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous...
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