Pavol Král

On the cardinalities of interval-valued fuzzy sets

Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+~[. We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice L^I (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed.

Implication Functions Generated Using Functions of One Variable

This chapter presents a survey of implication functions generated using an appropriate function of one variable including f-implications and g-implications introduced by Yager, h-generated implications introduced by Jayaram, h-implications and their generalizations introduced by Massanet and Torrens, I f and I g implications introduced by Smutna-Hliněna, and Biba.

Pre-orders and Orders Generated by Conjunctive Uninorms

This paper is devoted to studying of (pre-)orders of the unit interval generated by uninorms. We present properties of such generated pre-orders. Further we give a condition under which the generated relation is just a pre-order, i.e., under which it is not anti-symmetric. We present also a new type of uninorms, which is interesting from the point of view of generated pre-orders.

Aggregation operators preserving quasiconvexity

Quasiconvexity of a fuzzy set is the necessary and sufficient condition for its cuts to be convex. We study the class of those two variable aggregation operators that preserve quasiconvexity on a bounded lattice, i.e. A(@m,@n) is quasiconvex for quasiconvex lattice valued fuzzy sets @m, @n. The class of all such aggregation operators is characterized by a lattice identity that they have to fulfill. In case of a unit interval we show the construction of aggregation operators preserving quasiconvexity from a pair of real valued functions on the unit interval. As a consequence we get that the intersection of quasiconvex fuzzy sets is quasiconvex if and only if the intersection is based on the minimum triangular norm.

Choquet integral with respect to Łukasiewicz filters, and its modifications

In many decision problems a set of actions is evaluated with respect to a set of points of view, called criteria. This paper follows two aims - first to compare the so-called level-dependent Choquet integral introduced recently by Greco et al. with another transformation of Choquet integral, proposed by Havranova and Kalina. The other aim of this paper is to look for an appropriate utility function in a given setting. We illustrate our approach on a practical example, utilizing the level-dependent Choquet integral.

Generated Implications Revisited

In this paper we generalize f-generated fuzzy implications introduced by Yager. Further we generalize I f and \(I^g_N\) implications introduced by Smutna and RU-implications, studied by De Baets and Fodor, as well as (U,N)-implications. We study basic properties of these newly proposed fuzzy implications.

On the representation of cardinalities of interval-valued fuzzy sets: The valuation property

In the previous work, we developed an axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralaks axiomatic theory of the scalar cardinality of fuzzy sets. Cardinality was defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+~). We showed that the scalar cardinality of each interval-valued fuzzy set can be characterized using an appropriate mapping called a cardinality pattern. Moreover, we found some basic conditions under which the valuation property, the subadditivity property, the complementarity rule and the Cartesian product rule are satisfied using different cardinality patterns, t-norms, t-conorms and negations on the lattice L^I (the underlying lattice of interval-valued fuzzy set theory). This paper is the first in a series that further investigates the proposed theory, providing a description of cardinality patterns, t-norms, t-conorms and negations satisfying the properties mentioned above. This paper focuses on the valuation property.

Implications Generated by Triples of Monotone Functions

In this paper we deal with fuzzy implications generated via triples of monotone functions f,g,h. This idea has been presented for the first time at the IPMU 2012 conference, where we have introduced the generating formula and studied some special cases of these fuzzy implications. In our contribution we further develop this concept and study properties of generated fuzzy implications. More precisely,we study how some specific properties of generators f,g,h influence properties of the corresponding fuzzy implications.

Uninorms on Interval-Valued Fuzzy Sets

This paper is a kind of continuation of the paper by G. Deschrijver ‘Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory’, which was published in Information Sciences in 2013. In that paper he constructed uninorms whose neutral element is arbitrary of the type \({\mathbf e}=(e,e)\) and annihilator, \(\mathbf {a}\), is arbitrary point that is incomparable with \(\mathbf {e}\). In the present paper we intend to show what are all possibilities of the position of the pair \((\mathbf {e},\mathbf {a})\).

How to Calibrate a Questionnaire for Risk Measurement

Utility functions content parameters related to risk aversion coefficients which represent natural extensions of utility function properties. They measure how much utility we gain (or lose) as we add (or subtract) from our wealth. We set up these parameters for a person based on her/his answers to a questionnaire constructed to identify individual risk behavior. Calibration of such a questionnaire, and subsequently of utility functions, is based on an expected utility maximization of different alternatives of investment strategies. In the paper, we present questionnaire calibration methodology which we illustrate using absolute and relative risk aversion coefficients of two selected utility functions which have common, as well as different properties.