# Vedecké časopisy a ročenky vydávané na pôde SAV

## Tatra Mountains Mathematical Publications

Volume 1, 1992, No. 1

Content:

A physical example of quantum fuzzy sets, and the classical limit
Diederik Aerts 1), Thomas Durt 2), Bruno Van Bogaert 3)

 1) Theoretical Physics (TENA), Free University of Brussels; Pleinlaan 2; 1050 Brussels; BELGIUM. 2) Theoretical Physics (TENA), Free University of Brussels; Pleilaan 2; 1050 Brussels; BELGIUM. 3) Theoretical Physics (TENA), Free University of Brussels; Pleinlaan 2; 1050 Brussels; BELGIUM.

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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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 5-14.

Law of large numbers on D-poset of fuzzy sets
Ferdinand Chovanec 1), Mária Jurečková 2)

 1) Katedra matematiky, VA SNP; 031 19 Liptovský Mikuláš. 2) Katedra matematiky, Vojenská akadémia SNP; 031 19 Liptovský Mikuláš.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 15-18.

On a representation of observables in D-posets of fuzzy sets
Ferdinand Chovanec 1), František Kôpka 2)

 1) Katedra matematiky, VA SNP; 031 19 Liptovský Mikuláš. 2) Katedra matematiky, Vojenská akadémia; Demänovská cesta; 031 19 Liptovský Mikuláš.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 19-23.

Boolean methods in F-quantum spaces
Costas A. Drossos 1), M. Shakhatreh 2)

 1) Department of Mathematics, Faculty of Sciences; University of Patras; 261 10 Patras; GREECE. 2) Depatrment of Mathematics, University of Patras; Gr-261 10 Patras; GRECE.

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.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 25-37.

The compositional rule of inference with several relations
Robert Fullér 1), Brigitte Werners 2)

 1) Department of OR, Computer Center; Eötvös Loránd University; P.O.Box 157; H-1502 Budapest 112; HUNGARY. 2) Department of OR, RWTH Aachen; Templergraben 64; D-51000 Aachen; GERMANY.

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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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 39-44.

On the associativity of the product of modified real fuzzy numbers
Blahoslav Harman 1)

 1) Katedra matematiky VA SNP, 031 19 Liptovský Mikuláš.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 45-49.

Linguistic evidence and decision making
Stanis­ław Heilpern 1)

 1) Institute of Economic Cybernetics, Academy of Economics; Wroclaw; POLAND.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 51-55.

An optimum concept for fuzzified linear programming problems: a parametric approach
Francisco Herrera 1), Margit Kovács 2), J. L. Verdegay 3)

 1) Dpt. Computer Science and A. I., E.T.S. Ingeniería Informática; University of Granada; 18071-Granada;SPAIN. 2) Computer Centre, L. Eötvös University; P.O. Box 157; H-1502 Budapest 112; HUNGARY. 3) Computer Centre, L. Eötvös University; Budapest 112; P.O.Box 157, H-1502; HUNGARY.

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.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 57-64.

$g,p$-fuzzification of arithmetic operations
Tibor Keresztfalvi 1), Margit Kovács 2)

 1) Computer Centre, L. Eötvös University; P.O. Box 157; H-1502 Budapest 112; HUNGARY. 2) Computer Centre, L. Eötvös University; P.O. Box 157; H-1502 Budapest 112; HUNGARY.

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.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 65-71.

T-fuzzy observables
Anna Kolesárová 1), Beloslav Riečan 2)

 1) Katedra matematiky, ChTF STU; Radlinského 9; 812 37 Bratislava. 2) Matematický ústav SAV, Štefánikova 49; 814 73 Bratislava; Slovensko. riecan@fpv.umb.sk

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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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 73-82.

D-posets of fuzzy sets
František Kôpka 1)

 1) Katedra matematiky, Vojenská akadémia; Demänovská cesta; 031 19 Liptovský Mikuláš.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 83-87.

Some remarks on a system of semantical interpretation in natural languages
Mirosłava Kołowska 1)

 1) Institute of Mathematics, A. Mickiewicz University; Matejki 48/49; 60-769 Poznań; POLAND.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 89-91.

A representation of fuzzy quantum posets of type I, II
Le Ba Long 1)

 1) Khoa toán DHSP Hu{\^e}{}{{}'}, VIETNAM.

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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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 93-98.

Nontrivial example of an associative convolution
Peter Maličký 1)

 1) Department of Mathematics; Faculty of Natural Sciences; Matej Bel University;, Tajovského 40; SK--974-01 Banská Bystrica; SLOVAKIA. malicky@fpv.umb.sk

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 99-103.

Fuzzy sets and probability theory

 1) Katedra matematiky, Stavebná fakulta STU; Radlinského 11; 813 68 Bratislava.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 105-123.

States on soft fuzzy algebras - finite and countable additivity
Mirko Navara 1), Pavel Pták 2)

 1) Katedra matematiky, ČVUT FEL; Center for Machine Perception; Karlovo nám. 13; CZ-166 27 Praha 2; ČR. 2) Katedra matematiky Elektrotechnickej fakulty TU, Technická 2;166 27 Praha 6.

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].

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 125-134.

On a type of entropy of dynamical systems
Beloslav Riečan 1)

 1) Matematický ústav SAV, Štefánikova 49; 814 73 Bratislava; Slovensko. riecan@fpv.umb.sk

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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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 135-138.

Uncertainty measures of fuzzy propositions and their use in fuzzy inference
Jozef Šajda 1)

 1) Ústav teórie riadenia, a robotiky SAV; Dúbravská 9; 842 37 Bratislava.

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Tatra Mountains Mathematical Publications. Volume 1, 1992, No. 1: 141-150.