Eduardo Silva Palmeira

A new way to extend t-norms, t-conorms and negations

This work presents a method of extending t-norms, t-conorms and fuzzy negations to a lattice-valued setting by preserving the largest possible number of properties of these fuzzy connectives which are invariants under homomorphisms. Further, we also apply this method to extend De Morgan triples, automorphisms and n-dimensional t-norms.

Extension of fuzzy logic operators defined on bounded lattices via retractions

The primary goal of this paper is to present a method of extending t-norms, t-conorms and fuzzy negations from a sublattice M to the bounded lattice L by considering a more general version of the idea of the sublattice. In general terms, we consider M as a sublattice of the bounded lattice L, if M has the same lattice structure of the L equipped with the restriction of operations of L and is a subset of L. However, this latter condition may be relaxed without losing the essence of the usual definition of the sublattice. This is done through the use of retractions. Furthermore, the same idea is employed to extend t-subnorms and present some results related to extension and automorphism. Additionally, a formalization of a relaxed notion of De Morgan triple and its extension is provided.

Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions

Abstract In this work, we extend to the interval-valued setting the notion of overlap functions, presenting a method which makes use of interval-valued overlap functions for constructing OWA operators with interval-valued weights. Some properties of interval-valued overlap functions and the derived interval-valued OWA operators are analyzed. We specially focus on the homogeneity and migrativity properties.

On the extension of lattice-valued implications via retractions

The main goal of this paper is to apply the method of extension of fuzzy connectives proposed in our previous work for fuzzy implications valued on a bounded lattice. Also we discuss about which properties of implications are preserved by this method and we prove some results involving extension and automorphisms. Finally, we investigate the behavior of the extensions of two special classes of fuzzy implications, namely (S,N)-implications and R-implications.

A Comparison between K-Means, FCM and ckMeans Algorithms

Fuzzy C-Means, introduced by Jim Bezdek in 1981 is one of the earliest and most popular fuzzy clustering algorithms. However, in order to improve the hit rate or speed, over the years several modifications have been proposed. Among these we highlight the ckMeans algorithm proposed by us in 2010 which make a change in the way to calculate the center of the clusters of FCM. The idea is to use an auxiliary membership function of elements to those clusters that are essentially crisp and calculate the centroids following a similar process as done in K-Means algorithm but keeping the same procedures as in FCM in the rest of algorithm. In fact, this hybridization between FCM and K-Means motivated the name ckMeans for this variant of the FCM. In this article we apply K-Means, FCM and ckMeans algorithms in a validated database of mammograms with about a thousand elements and compare these three algorithms in terms of hit rate and number of each iterations and the computational processing time until the convergency of the system.

Some results on extension of lattice-valued QL-implications

BackgroundA very important issue in lattice theory is how to extend a given operator preserving its algebraic properties. For lattice-valued fuzzy operators framework, in 2008 Saminger-Platz presented a way to extend t-norms which was generalized by Palmeira et al. (2011) for t-norms, t-conorms, fuzzy negations and implications, considering the scenery provided by the (r,s)-sublattice.MethodsIn this paper we investigated how to extend QL-implications and which properties of it are preserved by the extension method via retractions (EMR).ResultsAs results, we proved that properties (LB), (RB), (CC1), (CC2), (CC3), (CC4), (L-NP), (EP) and (IP) are preserved by EMR.ConclusionsHowever, the extension method via retractions fails in preserving the important properties (NP), (OP), (IBL), (CP), (P) and (LEM).

Restricted Equivalence Function on L([0, 1])

It is know from the literature that interval-valued equivalence functions are not decomposable. In order to solve that problem and give a characterization for interval-valued restricted equivalence functions by means of aggregating interval fuzzy implication we consider an admissible order on the lattice L([0, 1]). Also, we discuss about some other properties of those operators.

A Generalization of a Characterization Theorem of Restricted Equivalence Functions

Fodor and Roubens’ equivalence functions (for short EF) are mapping normally used for making a comparison between images by means it can be used for measuring the similarity of images. So, having a suitable way to construct these functions is very important. In these sense, we present in this work a characterization theorem for restricted equivalence functions (a particular case of EF) using aggregation functions which is able to describe them from implications and vice-versa. We also present similar results for restricted dissimilarity functions and normal E e,N -functions.

Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets

This work was partially supported by the Brazilian Funding Agency CNPq under the Process 307781/2016-0, the Research Services of Universidad Publica de Navarra and by the research project TIN2016-77356-P from MINECO, AEI/FEDER, UE.

Pseudo-uninorms and Atanassov’s intuitionistic pseudo-uninorms

This paper considers the concept of pseudo-uninorm which is a natural generalization of the concept of pseudo t-norm for uninorms and presents the relationship between pseudo-uninorms on the unit interval and their natural extension according to Atanassovs intuitionistic framework, showing how to obtain, in a canonical manner, Atanassovs intuitionistic pseudo-uninorms from pseudo-uninorms. Moreover, we also show that the automorphisms on the unit interval and on L ∗ (the intuitionistic valued lattice) are in one-to-one correspondence and how automorphisms on L ∗ act on Atanassovs intuitionistic pseudo-uninorms.