Strict Triangular Norms Research Articles

Picture Fuzzy Interactional Bonferroni Mean Operators via Strict Triangular Norms and Applications to Multicriteria Decision Making

Based on the closed operational laws in picture fuzzy numbers and strict triangular norms, we extend the Bonferroni mean (BM) operator under the picture fuzzy environment to propose the picture fuzzy interactional Bonferroni mean (PFIBM), picture fuzzy interactional weighted Bonferroni mean (PFIWBM), and picture fuzzy interactional normalized weighted Bonferroni mean (PFINWBM) operators. We prove the monotonicity, idempotency, boundedness, and commutativity for the PFIBM and PFINWBM operators. We also establish a novel multi-criteria decision making (MCDM) method under the picture fuzzy environment by applying the PFINWBM operator. Furthermore, we apply our MCDM method to the enterprise resource planning (ERP) systems selection. The comparative results for our MCDM method induced by six classes of well-known triangular norms ensure that the best selection is always the same ERP system. Therefore, our MCDM method is effective for dealing with the picture fuzzy MCDM problems.

Remarks on two representations of strong negations and a connection between nilpotent and strict triangular norms

In this short communication, we will present a connection between two well-known representations of the strong negations (involutive negations). Namely, we will provide a necessary and sufficient condition for the equality of Trillas and Dombi forms of negations. We will also show a connection between the additive generators of nilpotent and strict triangular norms.

Convex hull of a fuzzy set and triangular norms

The t-norm fuzzy sets is a class of fuzzy sets equipped with a pair of algebraic operation, a notion of T-convexity and a metric, where all these notions are based on a strict triangular norm T. In this paper we introduce in this class a notion of T-convex hull of a fuzzy set. We prove theorem that binds the T-convex hull of an upper semicontinuous fuzzy set with the convex hull of a (crisp) set. We further show two applications of this result. First, we prove that the operation of forming T-convex hull behaves well with respect to algebraic operations. Second, we show an analogue of Shapley-Folkman theorem. Shapley-Folkman theorem is a well-known result that provides an upper bound on the distance between Minkowski sum of sets and the convex hull of this sum. In this paper we show that the distance between the sum of upper semicontinuous fuzzy subsets of a finite dimensional Euclidean space and the T-convex hull of this sum has an upper bound. As a consequence of this result, we present an iterative procedure for forming T-convex hull of an upper semicontinuous fuzzy set. As an application of the results regarding t-norm fuzzy sets we show an example from a game theoretic setting, where t-norm fuzzy sets were used to handle uncertainty in a repeated two player game.

Dominance on continuous Archimedean triangular norms and generalized Mulholland inequality

As a preceding result, it has been shown that the dominance relation is not transitive on the set of strict triangular norms. This result has been achieved thanks to new results on Mulholland inequality. Recently, Saminger-Platz, De Baets, and De Meyer have introduced the generalized Mulholland inequality which characterizes the dominance on all continuous Archimedean triangular norms in an analogous way as does Mulholland inequality on the strict triangular norms. Based on these new results, the present paper shows that the dominance relation is not transitive on the set of nilpotent triangular norms and, consequently, on the set of continuous Archimedean triangular norms. This result is achieved by introducing a new sufficient condition under which a given function solves the generalized Mulholland inequality and by showing that the set of the functions that solve the inequality is not closed with respect to compositions.

Some results on the convex combination of uninorms

It is well known that under some conditions, a convex combination of the drastic product TD and the strict triangular norm is still a triangular norm. Uninorms, as a special kind of generalization of both triangular norms and triangular conorms, have been widely used in fuzzy set theory. In this paper, the convex combination of uninorms is studied. Some properties of the class of uninorms with the underlying triangular norm TD are presented firstly. Then the convex combination of this class of uninorms is discussed and it is shown that only in some trivial cases, the convex combination is still a uninorm.

Implicit averaging functions

We propose a new type of weighted averaging functions given implicitly as a solution to an algebraic equation. The main ingredient is incorporation of weights performed by using a function different to the product. Strict triangular norms are one example of such functions. We show that the weights should be normalized in this case differently to the usual sum to one condition. We establish the conditions under which the implicit averages are well defined, are idempotent, monotone and bounded by the minimum and maximum of their arguments. We present many examples of implicit averages, some of which recover the existing families of functions while others define new families. In particular we provide new generalizations of power means, density based averages and weighted OWA functions.

Dominance on strict triangular norms and Mulholland inequality

This paper is devoted to the question of transitivity of the dominance relation defined on the set of strict triangular norms and to the connection between dominance and Mulholland inequality. A summary of the results reached in this field is given and a new result for the strict triangular norms of Family 19 is shown. Further, some parametric classes of solutions of Mulholland inequality, that do not comply with the Mulholland's condition, are presented and, finally, it is shown that the dominance relation is, in general, not transitive on the set of strict triangular norms.

On the Lipschitz property of strict triangular norms

This paper deals with Lipschitz triangular norms (t-norms). A partial answer to an open problem of Alsina, Frank and Schweizer is given with regard to strict t-norms with smooth additive generators. A new notion of local Lipschitz property for arbitrary t-norms is introduced. Some remarkable examples of non-Lipschitz continuous ones are provided.

Two-Dimensional Copulas as Important Binary Aggregation Operators

We introduce 2-copulas (copulas, shortly) and recent related research results. We present invariant copulas and their application in the theory of aggregation operators. Copulas are transformed by increasing bijections at the unit interval and discuss copula attractors. We also present results on the approximation of associative copulas by strict and nilpotent triangular norms.

Special Issue: Dedicated to INES 2005 and SISY 2005 Conferences

The current issue contains 12 papers presented at the <I>IEEE 9th International Conference on Intelligent Engineering Systems (INES 2005), Cruising on the Mediterranean Sea</I>, on September 16-19, 2005, and <I>3rd Serbian-Hungarian Joint Symposium on Intelligent Systems (SISY 2005), in Subotica, Serbia and Montenegro</I>, on August 31-September 1, 2005. The topics of the two conferences are very close to each other and regard Intelligent Systems both from practical and theoretical point of view. These successful conferences brought together active participants and joined researchers from several countries working on this very quickly developing, more and more important field. After a preliminary selection made by the section chairs and the International Program Committees, we have selected 12 papers to be published in extended form in the current Special Issue of the <I>Journal of Advanced Computational Intelligence and Intelligent Informatics</I>. We would like to express our thanks to our sponsors, the organizers and mainly to the participants, who made these scientific events possible. Also, we express our thanks to the Editors of the Journal of Advanced Computational Intelligence and Intelligent Informatics, for publishing this Special Issue. In the following we briefly describe each paper. A. Almeida, G. Marreiros present a model to support collaborative scheduling in complex dynamic manufacturing environments. This model considers the interaction between an Agent based Scheduling Module and a Group Decision Support Module. P. Baranyi, Z. Petres, P. L. Várkonyi, P. Korondi and Y. Yam study in their paper how the Tensor Product model transformation is capable of determining different types of convex hulls of the Linear Time Invariant models. The study is conducted through the example of the prototypical aeroelastic wing section. B. Bede, H. Nobuhara, J. Fodor and K. Hirota propose the study of the problem if usual sum and product can be substituted by max and product operations in defining approximation operators. In this sense max-product Shepard approximation operators are defined and studied. B. Benyó, P. Somogyi and B. Paláncz address the problem of classification of cerebral blood flow signals in order to identify the disorders of the cerebral circulation. The experimental results provided in the paper confirm the effectiveness of the proposed methods. J. Gáti and Gy. Kártyás propose a model based distance learning in the every day higher education practice in their contribution. They survey some important issues and methodological elements of virtual classrooms in comparison with demands for teaching procedures, programs, and materials. L. Horváth and I. J. Rudas propose a methodology for intelligent communication and change management for engineering modeling. This study motivated by increasing of importance of change management because of continuous product development. M. Maleković and M. Čubrilo describe in their contribution how to incorporate infatuation in multi-agent systems. Infatuation stands for the focusing on a single attractive or desirable characteristic of another agent and then considering the total agent as that one positive characteristic. E. Pap and M. Takács study two dimensional copulas as binary aggregation operators in their paper. Invariant copulas and an application of copulas in the theory of aggregation operators are discussed and a result on approximation of associative copulas by strict and nilpotent triangular norms is obtained. B. Reskó, Á. Csapó and P. Baranyi present in their contribution a visual cortex inspired cognitive model for contour and vertex detection. The contour detection and vertex extraction is performed by a vast network of simple units of computation simultaneously processing the visual data. The computational units are organized in a special structure, the Visual Feature Array. M. Takács addresses investigation of the problem of the approximate reasoning in the fuzzy systems, by reviewing a specific case, where the investigated structure is a real semi-ring with pseudo-operations. It is the investigation of special-type fuzzy sets, special g-generated t-norms and implications in approximate reasoning. J. K. Tar, I. J. Rudas and A. Rontó present in their paper a simple adaptive controller that creates only temporal and situation dependent system model. The temporal model can be built up and maintained step-by-step on the basis of slow elimination of fading information by the use of simple updating rules consisting of finite algebraic steps of lucid geometric interpretation. A. R. Várkonyi-Kóczy, A. Rövid and P. Várlaki present a new fuzzy based tone reproduction pre-processing algorithm which may help in developing the hardly or non-viewable features and content of the images making easier the further processing of it.

Residuated logics based on strict triangular norms with an involutive negation

AbstractIn general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t‐norms and their dual t‐conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t‐norms which are different from the product logic. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Building a class of fuzzy equivalence relations

In this paper, we propose a practical method, given a strict triangular norm with a convex additive generator, for deriving a fuzzy equivalence relation whose reflexivity condition generalizes Ruspini's definition of fuzzy partitions. The properties of the relations, their comparison, their transitivity, the construction of fuzzy equivalence relations on cartesian products are presented. A large part of the paper is devoted to applications with fuzzy partitions defined on the real line. Several examples, including the fairy-tale problem from De Cock and Kerre [On (un)suitable fuzzy relations to model approximate equality, Fuzzy Sets and Systems 133 (2) (2003) 137–153], the comparison of colored objects and comfort situations are proposed.

Pseudo-Additive Measures and Triangular-Norm-Based Conditioning

Conditioning in the framework of fuzzy measures (monotone normalized set functions vanishing in the empty set) is introduced. For every set i>B with non-null measure i>m(i>B) a conditional measure i>mi>B, based on a triangular norm i>T, is introduced. Universal conditioning preserving the lower semi-continuity is shown to be necessarily based on some strict triangular norm. Then also each conditional measure i>mi>B related to a pseudo-additive measure i>m is pseudo-additive. However, the pseudo-addition ⊕i>B operating on the measures i>mi>B is in general different from the pseudo-addition ⊕ operating on the measure i>m. Specific cases of universal conditioning preserving the pseudo-addition ⊕ are characterized. Classical probabilistic conditioning is shown to be a special case.

Characterization of Measures Based on Strict Triangular Norms

As a natural generalization of a measure space, Butnariu and Klement [4] introduced T-tribes of fuzzy sets with T-measures. They gave a complete characterization of T-measures for a Frank triangular norm T. Here we characterize T-measures with respect to non-Frank strict triangular norms. We show the specific roles of Frank triangular norms and a newly introduced family, nearly Frank triangular norms.