Vedecké časopisy a ročenky vydávané na pôde SAV

We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to fluctuations of the experimental apparatus, and not to an indetermination about the states of the physical entity, and is described by a varying parameter $varepsilon$. For zero value of the parameter (no fluctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal fluctuations), the example gives rise to a quantum fuzzy set, more precisely a spin-model. In between, we have a continuum of fuzzy situations, neither classical. nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.

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In this paper, we present a Boolean, point-free characterization of fuzzy observables, using Boolean-valued Dedekind cuts and the theory of Boolean powers. In the second part of the paper we study the links of Quantum spaces with the theory of orthospaces and its associated tolerance spaces. Finally in the third part using a soft Boolean algebra, we construct a Boolean model which incorporates all the previous ideas.

The compositional rule of inference with several relations, which is the mainly used inference rule in approximate reasoning, is considered in this paper. Stability results are given and exact computational formulae are provided.

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In this paper the optimality concept for $(g, p)$-fuzzified linear programming problems is studied. It is shown that this model can be solved by means of parametric linear programming problems. Moreover, some results about the $(g, p)$-fuzzified linear programming problem are obtained using the parametric linear programming problem.

The aim of this paper is to provide new results regarding the effective practical computation of $t$-norm-based arithmetic operations of $LR$ fuzzy numbers.

In the paper the notion of $T$-fuzzy observable is given and the properties of $T$-fuzzy observables are studied. The relation between $T$-fuzzy observables and random variables with values in the fuzzy real line, see, e.g., [E. P. Klement: Strong law of large numbers for random variables with values in the fuzzy real line, Communications of IFSA 187, Mathematics Chapter, 7 – 11], is shown, especially, the one-to-one correspondence between $T_∞$-fuzzy observables and finite fuzzy valued random variables is proved. The last section of the paper concerns with the calculus of $T$-fuzzy observables.

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Let $(Ω, M)$ be a fuzzy quantum poset of type I, II, or FQP of type I, II for short. For Boolean representations of fuzzy quantum spaces, see [M. Navara: Boolean representation of fuzzy quantum space, (to appear)]. By a representation of $(Ω, M)$ we mean a quantum logic $M$ (i.e., an orthocomplemented $σ$-orthocomplete orthomodular poset, see [V. S. Varadarajan: Geometry of Quantum Theory, Van Nostrand, New York, 1968] with a homomorphism $h:M\oversetonto\to\longleftrightarrowM$ such that for any state s on $M$ and any observable $\overline X$ on $M$ there is a state $\bar s$ on $M$ and observable $X$ on $M$ such that the following diagram commutes (where $B(\Bbb R)$ is the Borel $σ$-algebra of the real line $\Bbb R$).

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We bring a summary of recent results on states on soft fuzzy algebras (s. f. algebras) and point out some of their explicit consequences. We are mainly interested in the characterization of state spaces, the enlargements of s.f. algebras related to states, extensions of states, etc. A special attention is paid to the comparison of results for finitely additive states and for countable additive states. The basic technical tool for our investigation is a Boolean representation of s.f. algebras, which, we believe, might shed light on many questions of fuzzy logics, too (see [G. Cattaneo, M. L. Dallachiara, R. Guintini: Empirical semantics for fuzzy-intuitionistic quantum logic, to appear], [G. Cattaneo, M. L. Dallachiara, R. Guintini: Fuzzy intuitionistic quantum logics, to appear], [C. A. Drossos, M. Shakhatreh: Boolean methods in $F$-quantum spaces, Tatra Mountains Math. Publ. 1 (1992)], [A. Dvurečenskij: On a representation of observables in fuzzy measurable spaces, J. Math. Anal. Appl., to appear], [M. Navara: Algebraic approach to fuzzy quantum spaces, to appear], [J. Pykacz: Fuzzy set ideas in quantum logics, in: Proc. Conf. Quantum Logics, Gdańnsk, Poland, 1990].

This contribution has three aims. First we compare the concept by D. Dumitrescu [D. Dumitrescu: Measure preserving transformation and the entropy of fuzzy partition, in: 13th Linz seminar on Fuzzy set Theory (Linz 1991), pp. 25–27], [D. Dumitrescu: Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl. (to appear)] (see also [J. Rybárik: The entropy of $Q$–$F$-dynamical system, BUSEFAL 48 (1991), 24–26]) with that by P. Maličký and the author [P. Maličký, B. Riečan can: On the entropy of dynamical systems, in: Proc. Ergodic Theory and Related Topics II. (Georgenthal 1986), Teubner, Berlin, 1987, pp. 135–138]. Secondly we present two counting formulas for the entropy [P. Maličký, B. Riečan can: On the entropy of dynamical systems, in: Proc. Ergodic Theory and Related Topics II. (Georgenthal 1986), Teubner, Berlin, 1987, pp. 135–138]. Finally we present some remarks concerning the fuzzy entropy and especially we repeat the suggestion of P. Maličký to define a very close but different invariant for fuzzy dynamical systems.

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