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Descriptions of state spaces of orthomodular lattices (the hypergraph approach)

Mirko Navara — 1992

Mathematica Bohemica

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...

Considering uncertainty and dependence in Boolean, quantum and fuzzy logics

Mirko NavaraPavel Pták — 1998


A degree of probabilistic dependence is introduced in the classical logic using the Frank family of t -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 P -states, (resp. T -states) the consideration turned out to be fully analogous to (resp. considerably different from) the classical situation.

Validation sets in fuzzy logics

Rostislav HorčíkMirko Navara — 2002


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 S -fuzzy logics and extend them to the case of R -fuzzy logics.

Program for generating fuzzy logical operations and its use in mathematical proofs

Tomáš BartušekMirko Navara — 2002


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 [ 0 , 1 ] . 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 ( t -norms). It allows also to select these t -norms according to...

Automorphisms of concrete logics

Mirko NavaraJosef Tkadlec — 1991

Commentationes Mathematicae Universitatis Carolinae

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...

The σ-complete MV-algebras which have enough states

Antonio Di NolaMirko Navara — 2005

Colloquium Mathematicae

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 t -norm

Erich Peter KlementMirko Navara — 1997

Czechoslovak Mathematical Journal

We give a complete characterization of tribes with respect to the Łukasiewicz t -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 t -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 t -norms, e. g., for the product t -norm.

A Cantor-Bernstein theorem for σ -complete MV-algebras

Anna de SimoneDaniele MundiciMirko Navara — 2003

Czechoslovak Mathematical Journal

The Cantor-Bernstein theorem was extended to σ -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 σ -complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.

On interval homogeneous orthomodular lattices

Anna de SimoneMirko NavaraPavel Pták — 2001

Commentationes Mathematicae Universitatis Carolinae

An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is σ -complete and satisfies the following property: Whenever L is isomorphic to an interval, [ a , b ] , in L then L is isomorphic to each interval [ c , d ] with c a and d b (resp. the same condition as above only under the assumption that all elements a , b , c , d are central in L ). Let us denote by Inthom (resp. Inthom c ) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous...

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