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Program for generating fuzzy logical operations and its use in mathematical proofs

Tomáš Bartušek, Mirko Navara (2002)

Kybernetika

Fuzzy logic is one of the tools for management of uncertainty; it works with more than two values, usually with a continuous scale, the real interval [ 0 , 1 ] . Implementation restrictions in applications force us to use in fact a finite scale (finite chain) of truth degrees. In this paper, we study logical operations on finite chains, in particular conjunctions. We describe a computer program generating all finitely-valued fuzzy conjunctions ( t -norms). It allows also to select these t -norms according to...

Properties of fuzzy relations powers

Józef Drewniak, Barbara Pȩkala (2007)

Kybernetika

Properties of sup - * compositions of fuzzy relations were first examined in Goguen [8] and next discussed by many authors. Power sequence of fuzzy relations was mainly considered in the case of matrices of fuzzy relation on a finite set. We consider sup - * powers of fuzzy relations under diverse assumptions about * operation. At first, we remind fundamental properties of sup - * composition. Then, we introduce some manipulations on relation powers. Next, the closure and interior of fuzzy relations are examined....

Proto-metrizable fuzzy topological spaces

Francisco Gallego Lupiañez (1999)

Kybernetika

In this paper we define for fuzzy topological spaces a notion corresponding to proto-metrizable topological spaces. We obtain some properties of these fuzzy topological spaces, particularly we give relations with non-archimedean, and metrizable fuzzy topological spaces.

Quotient algebraic structures on the set of fuzzy numbers

Dorina Fechete, Ioan Fechete (2015)

Kybernetika

A. M. Bica has constructed in [6] two isomorphic Abelian groups, defined on quotient sets of the set of those unimodal fuzzy numbers which have strictly monotone and continuous sides. In this paper, we extend the results of above mentioned paper, to a larger class of fuzzy numbers, by adding the flat fuzzy numbers. Furthermore, we add the topological structure and we characterize the constructed quotient groups, by using the set of the continuous functions with bounded variation, defined on [ 0 , 1 ] .

Radon-Nikodym derivatives and conditioning in fuzzy measure theory.

Domenico Candeloro, Sabrina Pucci (1987)

Stochastica

In the last twenty years many papers have appeared dealing with fuzzy theory. In particular, fuzzy integration theory had its origin in the well-known Thesis of Sugeno [7]. More recently, some authors faced this topic by means of some binary operations (see for instance [3], [8] and references): a fuzzy measure must be additive with respect to one of them, an the integral is to define in a way, which is very similar to the construction of the Lebesgue integral. On the contrary, we are interested...

Relational Formal Characterization of Rough Sets

Adam Grabowski (2013)

Formalized Mathematics

The notion of a rough set, developed by Pawlak [10], is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of [6], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based models of rough...

Relative sets and rough sets

Amin Mousavi, Parviz Jabedar-Maralani (2001)

International Journal of Applied Mathematics and Computer Science

In this paper, by defining a pair of classical sets as a relative set, an extension of the classical set algebra which is a counterpart of Belnap's four-valued logic is achieved. Every relative set partitions all objects into four distinct regions corresponding to four truth-values of Belnap's logic. Like truth-values of Belnap's logic, relative sets have two orderings; one is an order of inclusion and the other is an order of knowledge or information. By defining a rough set as a pair of definable...

Relevance and redundancy in fuzzy classification systems.

Ana Del Amo, Daniel Gómez, Javier Montero, Gregory S. Biging (2001)

Mathware and Soft Computing

Fuzzy classification systems is defined in this paper as an aggregative model, in such a way that Ruspini classical definition of fuzzy partition appears as a particular case. Once a basic recursive model has been accepted, we then propose to analyze relevance and redundancy in order to allow the possibility of learning from previous experiences. All these concepts are applied to a real picture, showing that our approach allows to check quality of such a classification system.

Representation and construction of homogeneous and quasi-homogeneous n -ary aggregation functions

Yong Su, Radko Mesiar (2021)

Kybernetika

Homogeneity, as one type of invariantness, means that an aggregation function is invariant with respect to multiplication by a constant, and quasi-homogeneity, as a relaxed version, reflects the original output as well as the constant. In this paper, we characterize all homogeneous/quasi-homogeneous n -ary aggregation functions and present several methods to generate new homogeneous/quasi-homogeneous n -ary aggregation functions by aggregation of given ones.

Representation of uni-nullnorms and null-uninorms on bounded lattices

Yi-Qun Zhang, Ya-Ming Wang, Hua-Wen Liu (2024)

Kybernetika

In this paper, we present the representation for uni-nullnorms with disjunctive underlying uninorms on bounded lattices. It is shown that our method can cover the representation of nullnorms on bounded lattices and some of existing construction methods for uni-nullnorms on bounded lattices. Illustrative examples are presented simultaneously. In addition, the representation of null-uninorms with conjunctive underlying uninorms on bounded lattices is obtained dually.

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