Martin Kalina

T-evaluators and S-evaluators

The scalar evaluation of the aggregation of elements from a complete lattice via the operation meet (join) is compared with the aggregation of scalar evaluations of these elements via a t-norm (t-conorm). Based on this comparison, the notion of a T-evaluator (S-evaluator) is introduced and its basic properties are studied. The relationship between T-(S-) evaluators, fuzzy measures and generalized filters is investigated. An application of T-evaluators in fuzzy preference modelling is presented.

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.

Fuzzy smoothness and sequences of fuzzy smooth functions

Abstract This paper is a continuation of the work of Kalina (Tatra Mountains Math. Publ. 14 (1998) 153–159). It further investigates fuzzy smoothness of fuzzy continuous, but crisp, functions. Finally sequences of fuzzy smooth functions and their sums are considered.

Discrete and continuous recursive forms of OWA operators

Abstract We firstly present an evaluation problem for online shop based on gradually increasing number of inputs. Then we propose a model using Recursive OWA operator with constant orness/andness grade involved. Next, we analyze properties of discrete Recursive OWA operators, show their relationship with Pascal Triangle and further generalize this relationship to Γ function. For the continuous case, we propose a concept of self-similar ordered weighting functions (OWF) and analyze some properties of OWF. Using these concepts, we present the recursive aggregation method of continuous arguments under continuous OWF. We show a relationship of OWF with the Regular Increasing (RIM) quantifier. Furthermore, based on an isomorphism relation between discrete and continuous recursive OWA operators, Left Recursive OWA can be seen as the discrete form of continuous OWF.

Construction of uninorms on bounded lattices

Karaçal and Mesiar showed that it is possible to construct uninorms on an arbitrary bounded lattice. Particularly, they constructed the weakest and the strongest uninorm. In this paper another method for a construction of uninorms on bounded lattices is proposed. The cases of general bounded lattices as well as bounded lattices equipped with a special arithmetic operation, are discussed.

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 functions and capacities

Abstract Inspired by earlier results of Bayes and Benvenuti et al. [3] , we introduce and study some construction methods and transformations for capacities. First we introduce constructions of capacities based on n-ary aggregation functions and implications from fuzzy set theory, parameterized by vectors x ∈ [ 0 , 1 ] n . Later, implication based transforms of capacities, parameterized by subsets B ⊆ { 1 , 2 , . . . , n } , are given, exemplified and discussed.

Sugeno and Shilkret integrals, and T- and S-evaluators

At SISY 2008 conference we studied the Sugeno, Shilkret and Cho-quet integrals and some of their level-dependent modifications as TL-(SL-) evaluators. In this contribution we give an overview of possible level-dependent generalizations of Sugeno and Shilkret integrals. We show under which conditions these integrals and their generalizations are T- and/or S-evaluators for an arbitrary t-norm T and t-conorm S.

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.