Ferdinand Chovanec

Fuzzy quantum spaces and compatibility

Using ideas from fuzzy set theory, we generalize the notion of a quantum (probability) space introduced by Suppes to so-called fuzzy quantum spaces and we study the compatibility problem.

Difference Posets in the Quantum StructuresBackground

In this paper the conditions for D-posets to become orthoalgebras, orthomodularposets, orthomodular lattices, MV-algebras, and Boolean algebras are presented.Also some properties of observables are investigated. It is proved that any tworegular observables in an atomic σ-complete Boolean D-poset have a jointobservable.

On the topographic and optical properties of SiC/SiO2 surfaces

The roughness of the semiconductor surface substantially influences properties of the whole structure, especially when thin films are created. In our work 3C SiC, 4H SiC and Si/a-SiC:H/SiO2 structures treated by various oxidation a passivation procedures are studied by atomic force microscopy (AFM) and scanning tunnelling microscopy (STM). Surface roughness properties are studied by fractal geometry methods. The complexity of the analysed surface is sensitive to the oxidation and passivation steps and the proposed fractal complexity measure values enable quantification of the fine surface changes. We also determined the optical properties of oxidized and passivated samples by using visual modelling and stochastic optimization.

MV-Algebra Pasting

Z. Riečanová proved that every D-lattice is a set-theoretical union of MV-algebras. These MV-algebras are blocks in the D-lattice. There is a dual question: How can we construct a D-poset from a given collection of MV-algebras. To solve this problem we use the “pasting” technique. We define an admissible system of MV-algebras and we prove that the pasting of this system is a D-poset.

Ideals and Filters in D-Posets

Ideals and filters in D-posets are defined. Therole of idempotent elements and atoms in the theory ofideals and filters is studied.

Graphic representation of MV-algebra pastings

We deal with a construction of some difference posets via a method of a pasting of MV-algebras. We generalize Greechie diagrams used in MV-algebra pastings. We give necessary and sufficient conditions under which the resulting pasting of an admissible system MV-algebras is a lattice-ordered D-poset.

Fractal properties of MV-algebra pastings

Abstract We implement the fractal construction process to the MV-algebra pastings. Pasted MV-algebras generate special types of difference posets (D-posets). We compute the dimension of new fractal D-posets and show that the fractal structures discussed in this paper have a non-integer fractal dimension. Furthermore we observe an interesting agreement between the fractal dimension of the classical linear fractals (Cantor sets, Koch curves) and the dimension of fractal D-posets (Cantor fractal D-posets and Koch fractal D-posets).

Preface of the guest editors

The study of probabilistic models for random events, which are not simultaneously measurable or unsharp, requires new approaches to the modeling of uncertainty. The need for such an alternative theory comes back to the thirties of the twentieth century, since the classical (Kolmogorovian) probability theory could not explain events occurring in quantum physics. Nowadays, there are several scientific schools studying such events. Among utilized models, we can find orthomodular lattices (quantum logics), difference posets (effect algebras), multivalue algebras and fuzzy sets. This issue contains six papers devoted to topological and algebraic aspects of uncertainty modeling. These papers have been presented at the Ninth International Conference on Fuzzy Set Theory and Applications which was held in Liptovský Ján, Slovak Republic, 4–8 February 2008. In the first paper ‘‘Conditional states and independence in D-posets’’ by authors Chovanec Ferdinand, Drobná Eva, Kôpka František and Nánásiová Olga (Slovakia) conditional states on difference posets are introduced as a modification of Rényi’s definition of the conditional probability and utilizing the property of convexity of the space of states. An independence of elements with respect to a conditional state is investigated in D-posets. The second paper ‘‘Properties of max–*fuzzy relation equations’’ has been prepared by Drewniak Józef and Matusiewicz Zofia (Poland). In this article authors have extended the result of Zhang et al.(2006), who studied finite fuzzy relation equations with max–min and max– prod composition, such that the max–*composition is used for wide family of operations*. The main result of the third paper ‘‘On another view of an inverse of an interval matrix’’ by Lebedinska Julija (Latvia) is based on Rohn’s results in the field of interval matrixes. The problem of definition and determination of the fuzzy inverse matrix is studied. The fourth paper ‘‘Maps on a quantum logic’’ by Nánásiová Oľga and ValáškováĽubica (Slovakia) deals with functions of two variables on a quantum logic which correspond to measures of intersection, union and symmetric difference of two random events on a probability space. The fifth paper ‘‘Convex combinations of strict t-norms’’ by Petrı́k Milan (Czech Republic) is concerned with the class of strict t-norms whose multiplicative generators are determined by the first partial derivative along their zero borders. A subclass of strict t-norms where a counterexample cannot be found has been characterized and the remaining possibilities where such a counter-example could be found have been summarized. In the last paper ‘‘Variable-basis topological systems versus variable-basis topological spaces’’ by author Solovyov Sergey A. (Latvia), a variable-basis generalization of the notion of topological system of S. Vickers has F. Chovanec Department of Natural Sciences, Armed Forces Academy, 031 01 Liptovský Mikuláš, Slovak Republic

Erratum to: Conditional states and independence in D-posets

F. Koˆpka is deceased.The online version of the original article can be found underdoi:10.1007/s00500-009-0487-0.F. Chovanec (&) E. Drobna´Department of Natural Sciences, Academy of Armed Forces,P.O. Box 45, 031 01 Liptovsky´ Mikula´sˇ, Slovak Republice-mail: ferdinand.chovanec@aos.skE. Drobna´e-mail: eva.drobna@aos.skO. Na´na´siova´Department of Mathematics and Descriptive Geometry,Faculty of Civil Engineering, Slovak Technical University,Radlinske´ho 11, 813 68 Bratislava, Slovak Republice-mail: olga@vox.svf.stuba.skF. Chovanec E. Drobna´Mathematical Institute, Slovak Academy of Sciences,Sˇtefa´nikova 49, 814 73 Bratislava, Slovak Republic