Triangular Conorms Research Articles

Uninorms on bounded lattices with the underlying t-norms and t-conorms

Uninorms are a generalization of the concepts of triangular norms (t-norms) and triangular conorms (t-conorms) that admit a neutral element as any element in a bounded lattice. In this study, we discuss the characterization of uninorms on bounded lattices. We introduce some new constructions for uninorms on bounded lattices, which are derived from both t-norms and t-conorms. In particular, two idempotent uninorms on bounded lattices are obtained. We also provide some corresponding examples to help understand the structure of these new classes of uninorms. In addition, we provide an answer to the open problem proposed by Çaylı (2019) [14].

On constructing the largest and smallest uninorms on bounded lattices

Uninorms on the unit interval are a common extension of triangular norms (t-norms) and triangular conorms (t-conorms). As important aggregation operators, uninorms play a very important role in fuzzy logic and expert systems. Recently, several researchers have studied constructions of uninorms on more general bounded lattices. In particular, Çaylı (2019) gave two methods for constructing uninorms on a bounded lattice L with e∈L∖{0,1}, which is based on a t-norm Te on [0,e] and a t-conorms Se on [e,1] that satisfy strict boundary conditions. In this paper, we propose two new methods for constructing uninorms on bounded lattices. Our constructed uninorms are indeed the largest and the smallest among all uninorms on L that have the same restrictions Te and Se on [0,e] and, respectively, [e,1]. Moreover, our constructions does not require the boundary condition, and thus completely solved an open problem raised by Çaylı.

A new structure for uninorms on bounded lattices

This article investigates the construction of uninorms on bounded lattices. First, by using the fact that triangular norms (t-norms) and triangular conorms (t-conorms) on an arbitrary bounded lattice always exist, we introduce a new construction of uninorms on arbitrary bounded lattices with the neutral element. Then we illustrate how our new construction method is different from some existing methods for the construction of uninorms on bounded lattices. Some illustrative examples for the construction of uninorms on bounded lattices are provided.

Uninorms Having an Identity and an Annihilator on Bounded Lattices

Sınırlı kafesler üzerinde tanımlanan uninormlar, üçgensel normların ve üçgensel konormların önemli bir genelleştirmesidir ve bu operatörler, birimin sınırlı kafesin herhangi bir noktasında olmasına olanak sağlarlar. Bu çalışmada, sınırlı kafesler üzerinde uninormlar konusu üzerine çalışılmıştır. Sınırlı kafesler üzerinde bir birime ve sıfırlayana sahip uninormları karakterize etmek için bir metot önerildi ve bu şekildeki uninormların bazı temel özellikleri araştırıldı. Ayrıca, önerilen metotun uygulanabilirliğini göstermek için bir örnek verildi.

Analytical Structures of Takagi-Sugeno Fuzzy Two-Input Two-Output Proportional-Integral/Proportional-Derivative Controllers With Multiple Fuzzy Sets

In this paper, an attempt is made to generalize the analytical structures of Takagi-Sugeno (TS) fuzzy two-input two-output (TITO) proportional-integral (PI)/proportional-derivative (PD) controllers using multiple input fuzzy sets. Two models of fuzzy TITO PI/PD controllers are proposed based on two distinct control strategies. The inputs are fuzzified by multiple fuzzy sets with trapezoidal/triangular membership functions. The generalized rule base consists of nine control rules imbibing the complete control strategy and is closer in spirit to the original TS rule base. Algebraic product (AP) triangular norm, bounded sum (BS) triangular conorm, and center of gravity (CoG) defuzzifier are applied to derive the models. The models of the fuzzy TITO PI/PD controllers with multiple input fuzzy sets are (nonlinear) variable gain/structure controllers. Also, each output of the fuzzy controller is the sum of two nonlinear PI/PD controllers with variable gains. The gain variation and properties of the proposed controllers are studied. Two examples of nonlinear dynamic processes are considered to demonstrate the applicability of the proposed controllers.

Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

This paper deals with the simplest Takagi-Sugeno fuzzy two-term controllers. The analytical structures of the simplest fuzzy controllers were developed using a modified rule base. The number of tuning parameters of the controllers is reduced by introducing a novel rule base consisting of two rules. These controllers are termed as 'the simplest' since minimal (two) number of fuzzy sets is used for fuzzification. Algebraic product (AP)/minimum (min) triangular norm, bounded sum (BS)/maximum (max) triangular co-norm, different universes of discourses (UoDs) of inputs, and centre of gravity (CoG) defuzzification method are chosen to derive the mathematical models of the fuzzy controllers. The simplest fuzzy controller with modified rule base is equivalent to a (nonlinear) variable gain/structure PI/PD controller. The BIBO stability of the closed loop control system is investigated using the small gain theorem. The gain of the controller either varies or remains constant in different regions of the input plane. The gain variations and the computational burden of the fuzzy controllers are also studied. Two examples of nonlinear dynamical systems are considered to validate the developed models of the fuzzy controllers.

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.

A characterization of uninorms on bounded lattices by means of triangular norms and triangular conorms

ABSTRACTUninorms are generalizations of triangular norms and triangular conorms leaving the freedom for the neutral element to be an arbitrary element from a bounded lattice. In this paper, we study uninorms on bounded lattices and investigate their main characteristics. We also introduce the new construction methods for uninorms on bounded lattices with a neutral element based on the fact that the presence of triangular norms and triangular conorms. Furthermore, we assess and exemplify the differences between our constructions and the present approaches.

On the structure of uninorms on bounded lattices

Uninorms are a generalization of triangular norms and triangular conorms with a neutral element in the interior of a bounded lattice. In this study, we present two methods to construct uninorms from a given triangular norm and triangular conorm on bounded lattices under some constraints. As a by-product of these methods, we obtain new types of idempotent uninorms with a neutral element on bounded lattices. In addition, some illustrative examples for the constructions of uninorms on bounded lattices are provided.

Score Level Fusion of Multispectral Palmprint with Triangular Conorm

Biometricplay vigorous role in the authentication of user by using his/her physical body traits. Unimodal biometric system uses single body traits and multimodal systems use multiple body traits. Multimodal biometric system have overcome the disadvantages that has occurred in unimodal systems. In this paper we are fusing the different spectral bandsof palm print (Red, Green and Blue) using T-conorm operators like Hamacher, Frank, Probabilistic and Scheiwer & Sklar. Experiment Results suggest that Scheiwer & Sklar gives the best results. Experimental Results ascertain that the proposed approach for the score level fusion outperforms the state-of-art.

Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms

Abstract In fuzzy set theory, triangular norms (t-norm for short) and triangular co-norms (t-conorm for short) play a key role by providing generic models for intersection and union operations on fuzzy sets. Various continuous and discontinuous t-norms have been proposed by many authors. Despite variation in the t-norms, most of the well-known continuous t-norms are Archimedean (for example, Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family). An interesting family of non-Archimedean continuous t-norms was introduced by Dubois and Prade. This paper is an attemp to study a nonlinear optimization problem whose constraints are formed as a special system of fuzzy relational equations (FRE). In this type of constraint, FREs are defined with max-Dubois-Prade composition. Firstly, we investigate the resolution of the feasible solutions set. Then, some necessary and sufficient conditions are presented to determine the feasibility or infeasibility of the solutions set. Also, some procedures are introduced for simplifying the problem. Since the feasible solutions sets of FREs are non-convex, conventional nonlinear programming methods may not be directly employed to solve the problem. Therefore, in order to overcome this difficulty, a genetic algorithm (GA) is designed based on some theoretical properties of the problem. It is shown that the proposed algorithm preserves the feasibility of new generated solutions. Moreover, a method is presented to generate feasible max-Dubois-Prade FREs as test problems. These test problems are used to evaluate the performance of our algorithm. Finally, the algorithm are compared with some related works. The obtained results confirm the high performance of the proposed algorithm in solving such nonlinear problems.

Analysis of additive generators of fuzzy operations represented by rational functions

This article presents an approach for determining additive generators of commutative and associative operations. Its applicability for finding generators of triangular norms and conorms is shown. Conditions for the parameters of increasing generators that generate triangular conorms in the class of rational functions are determined.

Edge Detection on DICOM Image using Triangular Norms in Type-2 Fuzzy

In image processing, edge detection is an important venture. Fuzzy logic plays a vital role in image processing to deal with lacking in quality of an image or imprecise in nature. This present study contributes an authentic method of fuzzy edge detection through image segmentation. Gradient of the image is done by triangular norms to extract the information. Triangular norms (T norms) and triangular conorms (T conorms) are specialized in dealing uncertainty. Therefore triangular norms are chosen with minimum and maximum operators for the purpose of morphological operations. Also, mathematical properties of aggregation operator to represent the role of morphological operations using Triangular Interval Type-2 Fuzzy Yager Weighted Geometric (TIT2FYWG) and Triangular Interval Type-2 Fuzzy Yager Weighted Arithmetic (TIT2FYWA) operators are derived. These properties represent the components of image processing. Here Edge detection is done for DICOM image by converting into 2D gray scale image, using Type-2 fuzzy MATLAB and which is the novelty of this work.

An Analytical Structure of the Simplest Takagi-Sugeno TITO PI/PD Controller

An analytical structure of the simplest Takagi-Sugeno (TS) fuzzy two-input two-output (TITO) PI/PD controller is presented in this paper. A mathematical model is derived using two fuzzy sets on each input, five control rules, algebraic product (AP) triangular norm, bounded sum (BS) triangular co-norm, different universes of discourse (UoD) for different inputs, and center of gravity (CoG) defuzzifier. The controller is termed as ‘the simplest’ since minimum number (two) of fuzzy sets is employed on each of the inputs. The structure of the controller reveals that it is a (nonlinear) variable gain/structure controller. Also, it is shown that each output of the simplest TS fuzzy TITO PI/PD controller is the sum of two individual PI/PD controllers with nonlinear gains. Simulation results are presented to validate the controller model.

Operations on Hesitant Linguistic terms sets Induced By Archimedean Triangular Norms And Conorms

The aim of the paper is to discuss some new operations on hesitant fuzzy linguistic terms sets based on Archimedean t-norm and t-conorm. The advantage is that the operations on hesitant fuzzy linguistic terms sets are closed, by studying propositions of the operations on hesitant fuzzy linguistic terms sets, scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators are proposed. An example is presented to illustrate the practicality of the four well-known scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators, which are also compared with the symbolic aggregation-based method in the example, results show that scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators can be applied to fuse hesitant fuzzy linguistic terms sets.

On transformations from semi-three-way decision spaces to three-way decision spaces based on triangular norms and triangular conorms

Recently, three-way decisions on three-way decision spaces based on partially ordered sets have been proposed by the axiomatic definitions. And, in the theory of three-way decision spaces, the decision evaluation function (which is defined by the so-called minimum element axiom, monotonicity axiom and complement axiom) is a vital notion and plays pretty important role in practical decision-making problems. However, there are many meaningful handy functions which can not become decision evaluation functions because they only satisfy minimum element axiom and monotonicity axiom in general. Therefore, there arises one discussion in which the author introduced the concept of semi-decision evaluation function without complement axiom lately. And three-way decisions based on semi-three-way decision spaces are proposed. This paper continues this research topic and gives several transformation methods from semi-three-way decision spaces to three-way decision spaces based on triangular norms and triangular conorms. Based on them, decision-makers can obtain more useful decision evaluation functions thereby having more choices in realistic decision-making problems. In addition, as an application, we analyse a reality example of an evaluation problem of credit card applicants by using the results obtained in this paper.

A Novel Picture Fuzzy Linguistic Aggregation Operator and Its Application to Group Decision-making

The intuitionistic fuzzy set (IFS) is an effective tool to express uncertain and incomplete cognitive information with the membership, non-membership, and hesitance degrees. The picture fuzzy set (PFS) is an extension of IFS, and it can contain more information than IFS. Linguistic variables (LVs) can easily describe the complex cognitive information provided by the decision makers. So it is necessary to combine the PFS with the linguistic term and define the picture fuzzy linguistic set (PFLS), then extend Archimedean triangular norms (t-norms) and triangular conorms (s-norms), which are the generalizations of a number of other t-norms and s-norms, such as Einstein, Hamacher, and Frank s-norms and t-norms, to process PFLS. By combining the PFS and the linguistic term, we define the picture fuzzy linguistic set (PFLS) and the operations of the picture fuzzy linguistic numbers (PFLNs). Further, we develop the Archimedean picture fuzzy linguistic weighted arithmetic averaging (A-PFLWAA) operator and discuss a number of properties and special cases of this operator. Besides, we propose an approach to deal with multiple attributes group decision-making (MAGDM) problem on the basis of the developed A-PFLWAA operator. An approach to solve MAGDM problem is proposed on the basis of the A-PFLWAA operator, and the validity and reliability of the proposed method are proven by an illustrative example. The PFLNs are the generalizations of the intuitionistic linguistic numbers (ILNs), and they contain more information than ILNs. In addition, our method cannot only be utilized to solve the problems with ILNs but also be used to deal with the problems with PFLNs, and it is a generalization of a number of the existing methods.

Aggregation of diverse types of fuzzy orders for decision making problems

In this paper, conditions for n-argument functions to preserve diverse types of fuzzy orders during aggregation process are presented. Namely, the following fuzzy orders are studied: total preorder, total order, strict total order, partial preorder, partial order, strict partial order, interval order or semiorder and their preservation in aggregation process along with other connections between input fuzzy relations and the aggregated one are presented. Moreover, dependencies between diverse classes and also among the given class of fuzzy orders are presented. The considered fuzzy orders are compound properties depending in their definitions on arbitrary binary operations on [0, 1], including fuzzy conjunctions and disjunctions (with special cases of triangular norms and triangular conorms). Finally, an application of the obtained result is presented by the given algorithm.

Idempotent nullnorms on bounded lattices

Nullnorms are generalizations of triangular norms and triangular conorms with a zero element to be an arbitrary point from a bounded lattice. In this paper, we study and discuss the existence of idempotent nullnorms on bounded lattices. Considering an arbitrary distributive bounded lattice L, we show that there exists a unique idempotent nullnorm on L. We prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice. Furthermore, we propose a construction method to obtain idempotent nullnorms on a bounded lattice L with an additional constraint on a for the given zero element a ∈ L\\{0, 1}.

On a new class of t-norms and t-conorms on bounded lattices

The ordinal sum of triangular norms on unit interval has been used to construct other triangular norms. But, in general, ordinal sum construction of triangular norms and triangular conorms may not work on an arbitrary bounded lattice. In this study, we focus on ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice. Moreover, we show a complete generalization of our ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice. And some illustrative examples are added for clarity.