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The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time. [ABSTRACT FROM AUTHOR]

This third and last part of a series of position papers on triangular norms (for Parts I and II see (E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press; E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper II: general constructions and parameterized families, submitted for publication) presents the representation of continuous Archimedean t-norms by means of additive generators, and the representation of continuous t-norms by means of ordinal sums with Archimedean summands, both with full proofs. Finally some consequences of these representation theorems in the context of comparison and convergence of continuous t-norms, and of the determination of continuous t-norms by their diagonal sections are mentioned. [Copyright &y& Elsevier]

This second part (out of three) of a series of position papers on triangular norms (for Part I see Triangular norms. Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press) deals with general construction methods based on additive and multiplicative generators, and on ordinal sums. Also included are some constructions leading to non-continuous t-norms, and a presentation of some distinguished families of t-norms. [Copyright &y& Elsevier]

This brief paper addresses problems that were raised in the paper (Fuzzy Sets and Systems, this issue) in connection with modelling fuzzy equality and similarity. In addition to specific comments to the cited paper, we also point out the more fundamental questions underlying this discussion. [Copyright &y& Elsevier]

In this position paper, we shall undertake a critical scrutiny of the decision analysis (DA) paradigm on the basis of experience we gained from working on the bullwhip effect. We had some success with sorting out the complexities of the bullwhip by using fuzzy numbers in the bullwhip models, and we found out that this points the way to a more viable approach to sort out complex decision problems. As a positive criticism of the DA paradigm we have collected the insights we have gained in an agenda for soft decision analysis (SDA), which we hasten to add is not complete but—hopefully—an opening for focused research and continuing development. [Copyright &y& Elsevier]

Abstract: The purpose of this note is to correct two convergence theorems for the generalized fuzzy integrals given in Fang [On the convergence theorems of generalized fuzzy integral sequence, Fuzzy Sets and Systems 124 (2001) 117–123]. [Copyright &y& Elsevier]

We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given. [Copyright &y& Elsevier]

Abstract: In this paper we deal with the construction of global fuzzy preference structures from two-parameter single-criterion fuzzy outranking relations proposed in [10] and which are based on the aggregation of outranking indices associated with -cuts. We consider two ways of constructing global fuzzy preference structures: (i) aggregate the two-parameter single-criterion fuzzy outranking relations and then construct the global fuzzy preference structures and (ii) construct global fuzzy preference structures and then aggregate the single-criterion fuzzy preference, indifference and incomparability relations. We also investigate the case where these two ways of proceeding lead to the same results. [Copyright &y& Elsevier]

Abstract: We discuss various epistemological, foundational and practical aspects concerning the treatment of uncertainty, expressing the opinion that the approach followed in de Cooman''s paper is neither useful nor convincing. [Copyright &y& Elsevier]

Abstract: Some Virtues of possibilistic previsions introduced by Gert de Cooman in the paper A behavioural model for vague probability assesments are pointed out from the application point of view. A member of questions concerning extensions of the possibilistic previsions are posed. [Copyright &y& Elsevier]

Abstract: In reply to Dubois and colleagues, here it is discussed the relationship between the notion of intuitionistic fuzzy set as proposed by the author in 1983 and the notion proposed in 1984 by Takeuti and Titani under the same name. It is demonstrated also the main direction of research, which does not (and in some cases, cannot) have analogues in fuzzy sets theory. [Copyright &y& Elsevier]

Identification of a multi-input/single-output system enables the creation of a mathematical model that as accurately as possible represents the input-output relation induced by a physical system. When there is little information about the physical laws related to the system or when the system is too complex, methods such as parametric identification are used to define the system model. In that case, preliminary assumptions can be made about the system leading to a parametric aggregation function based model. This parametric model may be learned by estimating the parameters of this aggregation function from a representative set of inputs/outputs. A major difficulty is to design a model that is relatively simple yet precise enough to meet the user's needs. Linear models are commonly used because they meet these two contradictory constraints. However, use of a linear model is often at the expense of the accuracy of input-output relationship description. In a recent paper, a new modeling approach was proposed, under the name of macsum modeling [31] , which aims at replacing the linear model concept by a set of linear models. The obtained aggregation function leads to an interval-valued output. This represents the lack of accuracy of the model to predict the system output when the inputs are known. An interesting feature of this model is that it is ruled by a single precise parametric vector whereas the output is imprecise. Moreover, the vector dimension is equal to that of the input space. In this paper, we propose a method to learn such a model so that it best reflects the input-output relationship of the considered system. This method is based, as in the case of linear modeling, on a regression method. This is particularly new approach because the macsum aggregation is based on a Choquet integral and very few authors have proposed learning of an aggregation function based on the Choquet integral. [ABSTRACT FROM AUTHOR]

In a recent paper by the authors, Jensen's inequality for Choquet integral was given, and a wrong assertion—"Jensen's inequality does not hold for asymmetric Choquet integral" was made. This paper can be viewed as a continuation of the previous one, Jensen's inequality for asymmetric Choquet integral is proved, the error is corrected. As its generalization, Jensen's inequality for generalized asymmetric Choquet integral is obtained. [ABSTRACT FROM AUTHOR]

This paper discusses the finite-time stabilization of a class of discontinuous fuzzy neutral-type neural networks with multiple time-varying delays. For the purpose of achieving this stabilization, first of all, by using the differential inclusions and set-valued mapping, the differential inclusion system is established in order to cope with the discontinuities. Then, by means of the introduced sign function, delayed discontinuous control strategies are given. By constructing a modified Lyapunov-Krasovskii functional concerning with the mixed delays, delay-dependent criteria formulated by algebraic inequalities are derived for guaranteeing the finite-time stabilization. Moreover, the settling time is estimated. Finally, numerical examples are carried out to verify the effectiveness of the established results. Since this kind of stabilization analysis of such neural system in this work has not been given much attention, the delay-dependent criteria derived in this paper can be regarded as a leading stabilization result for fuzzy neutral-type neural networks with D operator including multiple time and multiple neutral delays. [ABSTRACT FROM AUTHOR]

Interpretability in intelligent models becomes a challenge in academic research and approaches that facilitate understanding the responses obtained in models based on artificial intelligence and machine learning. This paper presents a new logical fuzzy neuron based on the concept of null-uninorm, thus called null-unineuron to compose the architecture of an evolving neuro-fuzzy model. This new structural neuron can extract advanced fuzzy rules allowing AND and OR-connections of antecedents to better interpret and understand the analyzed problem. This three-layer model uses an evolving weighted fuzzification approach based on incremental data partitioning concepts for knowledge extraction through null-unineurons whose training procedure suits the classification of binary and multiclass patterns in an online and incremental way. The weights integrated in the evolving data partitioning algorithm belong to feature importance levels and achieve an automatic shrinkage of distance calculations along unimportant input directions (features), which in turn accounts for a soft dimension reduction and the likelihood to decrease over-fitting. The new architecture proposed in this model was, subject to pattern classification tests, being more efficient compared to related (evolving) neuro-fuzzy models in the literature. Finally, experiments on various real-world data sets proved that the evolving neuro-fuzzy model proposed in this paper can act in a simplified way in the extraction of knowledge from data while providing answers with a high degree of accuracy for pattern classification problems. [ABSTRACT FROM AUTHOR]

Möbius transforms of capacities or games were considered as a tool for extensions of capacities to particular aggregation functions in several papers. This is, for example, the case of the Lovász extension coinciding with the Choquet integral, or the Owen extension that is also known as multilinear extension. In this paper, a much deeper study of the links between the Möbius transforms of real-valued functions defined on finite bounded posets and aggregation functions is performed. We introduce a Möbius product of any two real-valued functions defined on an arbitrary finite bounded poset, and then, fixing the poset (2 N , ⊆) , N = { 1 , ... , n } , we propose and discuss a construction method for n -ary aggregation functions based on the Möbius product of any capacity on N and a real-valued function g x , x ∈ [ 0 , 1 ] n , defined on 2 N and determined by some appropriate n -ary aggregation function. We provide some necessary and some sufficient conditions for the introduced construction to yield an aggregation function for any capacity. For the binary case, a complete characterization of conditions under which our approach results in an aggregation function for each capacity m is given. [ABSTRACT FROM AUTHOR]

Uncertainty theory was founded by Baoding Liu for modeling belief degrees about human uncertainty. And chance theory was pioneered by Yuhan Liu based on probability theory and uncertainty theory. In many cases, imprecise risk situation in subjective or objective setting and human uncertainty simultaneously appear in a complex system. To measure uncertain random events in this system, this paper combines uncertainty space with convex non-additive probability space into a new kind of two-dimensional chance space called U-C chance space. Moreover, a new framework for uncertain random variables combining uncertainty theory with convex non-additive probability theory is provided. For applications of this new framework, it can be applied to characterize some kinds of phenomena which possess the characteristics of both imprecise risk situations and belief degrees about human uncertainty in situations of great events, such as financial crisis, major natural disaster, etc. The main contribution of this paper is to derive the types of Kolmogorov and Marcinkiewicz-Zygmund LLNs for uncertain random variables satisfying some conditions under U-C chance space, where the convex non-additive probability is totally monotone. Furthermore, several examples are stated and explained. [ABSTRACT FROM AUTHOR]

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results. [ABSTRACT FROM AUTHOR]

Three basic issues of granular computing are construction or definition of granules, measures of granules, and computation or reasoning with granules. This paper reviews the main theories of granular computing and introduces the definition of spatial granules. A granule is composed of one or more atomic granules. The rationality of this definition is explained from the four aspects: simplicity, applicability, measurability and visualization. A one-to-one correspondence is established between the granules and the points in the unit hypercube, and the coarsening and refining of the granules are the descending and ascending dimensions of the points, respectively. The weak fuzzy tolerance relation and weak fuzzy equivalence relation are defined so as to study on all fuzzy binary relations. The notion of layer granularity/fineness is introduced and each granule can be easily denoted by two numbers, which can be used to pre-process macro knowledge space and greatly improve the search speed. This paper also discusses the main properties of granules including the necessary and sufficient conditions of coarse-fine relation and the main principles of granular space. • Introduce the definition of spatial granules and explain its rationality. • Define coarse-fine relation and all the granules can be hierarchicalized. • Adopt a one-to-one mapping between the granules and the points. • Discuss the main properties of spatial granules. • Layer granularity/fineness is used to pre-process macro knowledge space. [ABSTRACT FROM AUTHOR]

The algebraic theory of quasi-Nelson logic, a non-involutive generalization of Nelson's constructive logic with strong negation, has been shown to be surprisingly rich in a series of recent papers. In the present paper we bring quasi-Nelson logic into the fuzzy setting by adding the prelinearity axiom to it. We observe that the resulting system is an extension of the well-known Weak Nilpotent Minimum logic, as well as a rotation logic in the sense of recent work by P. Aglianò and S. Ugolini. We characterize the algebraic models of prelinear quasi-Nelson logic as twist-structures over Gödel algebras endowed with a nucleus operator and use the insight thus gained to look at subvarieties corresponding to extensions of well-known fuzzy systems. Our study of the quasi-Nelson negation in a prelinear setting also allows us to show that the variety of prelinear quasi-Nelson algebras is generated by a single standard algebra, thus obtaining a single chain completeness theorem for the logic. [ABSTRACT FROM AUTHOR]

Uncertainty extensively exists in data and knowledge intensive applications, in which fuzzy information processing plays a crucial role. Fuzzy sets have been extensively used to enhance various database models for managing fuzzy data or flexibly querying crisp data. This has resulted in numerous contributions in this research area. This paper pays attention to three crucial issues in fuzzy techniques for data management: modeling fuzzy data , querying fuzzy data , and fuzzy queries over crisp data , and provides a full up-to-date survey on the current state of the art in fuzzy data modeling and querying. The paper identifies fuzzy conceptual data models, fuzzy (relational and object-oriented) database models and fuzzy XML model as well as the relationships among these fuzzy data models. For each type of fuzzy data models, the paper summarizes its query processing. The paper also reviews fuzzy querying over classical data models. In addition to providing a generic overview of the approaches for fuzzy data modeling and querying, this survey paper serves for identifying possible research opportunities in the area of fuzzy data processing. [ABSTRACT FROM AUTHOR]

Graph theory has been a useful tool for logic programming in many aspects. In this paper, we propose an equivalent representation of multi-adjoint logic programs using hypergraphs, which are a generalization of classical graphs that allows the use of hypergraph theory in logic programming. Specifically, this representation has been considered in this paper to increase the level and flexibility of different termination results of the computation of the least model of fuzzy logic programs via the immediate consequence operator. Consequently, the least model of more general and versatile fuzzy logic programs can be obtained after finitely many iterations, although infinite programs or programs with loops and general aggregators will be considered. [ABSTRACT FROM AUTHOR]

In the paper Jónás et al. (2022) [1] we asserted a sufficient condition for the concavity (convexity) of the tau function, which is a generator function-based parametric mapping. In our original definition, the generators of the tau function are all the additive generators of the strict triangular norms and strict triangular conorms. We realized that with the original definition, our assertion regarding the concavity (convexity) of the tau function is not valid for all its generators. In this erratum, we present a corrected definition for the tau function such that with this new definition the original assertion regarding the concavity (convexity) of the tau function is valid. With the new definition for the tau function, on the one hand, we narrow the class of its generators, but on the other hand, this class still remains quite versatile and all the other results of the referenced paper remain valid. [ABSTRACT FROM AUTHOR]

There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral and commutative, then Q -valued domains are Q -enriched, and every Q -enriched domain is sober in its Scott Q -valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of Q -enriched dcpos and Q -enriched domains in the general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right subdistributive quasi-magmas on Q in the sense of the category Cat (Q). It is important to point out that our quasi-magmas on Q are in tune with the «intersection axiom» of Q -enriched topologies. When Q is involutive, each Q -enriched domain becomes sober in its Q -enriched Scott topology. This paper also offers a perspective to apply Q -enriched dcpos to quantale computation. [ABSTRACT FROM AUTHOR]

The research purpose of this paper is to explore some new fuzzy rough set models in the fuzzy β -covering group approximation space (F β CGAS) and study the related properties, thereby provide new thinking directions for the theoretical development and application expansion of fuzzy rough set models. Firstly, in the fuzzy β -covering approximation space (F β CAS), in light of some existing fuzzy covering rough set (FCRS) models which do not satisfy the inclusion relations between the fuzzy upper and lower approximations, we define the concepts of parameterized fuzzy β -neighborhoods and parameterized fuzzy complementary β -neighborhoods to eliminate part of the noise data and establish four types of parameterized FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. Meanwhile, the properties of the models are discussed. Further, in the F β CGAS, we define the concepts of the parameterized pessimistic (optimistic) fuzzy β -fitting neighborhoods and the parameterized pessimistic (optimistic) fuzzy complementary β -fitting neighborhoods and construct the parameterized pessimistic and optimistic FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. At the same time, we likewise explore the properties of the models. Besides, we further use the newly proposed parameterized fuzzy neighborhoods to study the variable precision multi-granulation fuzzy decision-theoretic rough set models in the F β CGAS. Finally, for all the models mentioned above, we recommend them for solving the criteria weight determination problem, ranking problem, optimal selection problem and qualitative classification problem in multi-criteria decision-making. Meanwhile, we complete the relationships of all the new models mentioned in this paper and discuss them. [ABSTRACT FROM AUTHOR]

We know from a previous paper that the reticulation of a coherent quantale A is a bounded distributive lattice L (A) whose prime spectrum is homeomorphic to m -prime spectrum of A. This paper studies how the reticulation can be used for transferring some properties of bounded distributive lattices to quantales and vice versa. We shall illustrate this thesis by proving several results on the pure and w –pure elements of the quantale A by means of the reticulation L (A). In particular, we shall investigate how the properties of σ –ideals of L (A) can be transferred to pure and w –pure elements of A. Then the obtained transfer properties are used to prove new algebraic and topological results and characterization theorems for some important classes of quantales: normal quantales, mp –quantales, PF –quantales, purified quantales and PP –quantales. [ABSTRACT FROM AUTHOR]

There are two cases of the (conditional) distributivity of semi-t-operators over S -uninorms that have not been discussed by Fang and Hu in 2019. This paper further characterizes the necessary condition for the left (resp. right) distributivity of semi-t-operators over S -uninorms in these two cases. The sufficient conditions for the left (resp. right) distributivity of semi-t-operators over S -uninorms are also discussed. The outcome of one case is different from the conclusion obtained by Fang and Hu, whereby it is impossible to obtain the necessary and sufficient conditions for semi-t-operators distributed over S -uninorms as usual. In particular, this paper points out that the distributivity of semi-t-operators over S -uninorms is equivalent to their conditional distributivity over S -uninorms under the two cases mentioned above. Some examples are also presented to support our conclusions. [ABSTRACT FROM AUTHOR]

The characterization of (S , N) -implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [ 0 , 1 ] 2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [ 0 , 1 ] 2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of (S , N) -implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [ 0 , 1 ] 2 or a strict t-conorm. [ABSTRACT FROM AUTHOR]

The research on fuzzy implication functions has exponentially grown in the last decades becoming one of the main topics studied in the Fuzzy Logic community. These efforts have led to significant advances on the theoretical (mainly) and applied aspects, but the overwhelming number of papers hinder the tracking of the new results and the problems that remain unsolved. The goal of this paper is to determine the valuable practices in the research on classes of fuzzy implication functions and the desirable characteristics that a paper on this topic should have to be useful for a reader. Clearly adopting these features can accelerate the pace of the future research and be constructive for new researchers. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method to judge the degree of the relationship closeness between system input variables and theoretical output through independence test, and apply this method to construct a hierarchical interval type-2 Takagi-Sugeno-Kang (TSK) fuzzy inference system. Interval type-2 TSK fuzzy inference systems have been applied in many fields. An interval type-2 fuzzy inference system, which is a special case of generalized type-2 fuzzy inference systems, can reduce the computational complexity of type-reduction and save computational cost. An hierarchical fuzzy system can further reduce the complexity of systems. However, it is found that, with the increase in data dimensions, the computational complexity and error of the system will increase. The system accuracy of a hierarchical fuzzy system is also found related to the order of its input variables. Variable selection can select meaningful input variables to decrease input space dimensions, and sort the input variables in some way. This paper proposes a variable selection and sorting method based on independence test. Firstly, input variables data and theoretical output data are classified and contingency tables are established. Chi-square statistics of the contingency tables are calculated for independence test, and input variables that are not independent of theoretical output are selected. Then, the total inertia of the contingency tables is calculated to represent the degree of the relationship closeness between input variables and theoretical output. According to the total inertia, the input variables are sorted and entered the system in order. The proposed method and the model built by the method are tested by dealing with regression problems. By comparing the proposed method on several types of systems, the hierarchical interval type-2 TSK inference system shows a better performance. Then the model is compared with other advanced fuzzy models and non-fuzzy models. Experiments show that its performance is better than the rest. [ABSTRACT FROM AUTHOR]

In data sets some attributes may have higher or lower importance. One of the tools used for data analysis of such datasets are subspace neuro-fuzzy systems. They elaborate fuzzy rules to describe data sets. In subspace neuro-fuzzy systems fuzzy rules exist in subspaces defined with subsets of attributes. In the paper we propose a novel fuzzy biclustering algorithm that groups both objects and attributes in fuzzy clusters. In that way we create a subspace fuzzy rule base for a subspace fuzzy system. The paper is accompanied with numerical examples that show this approach can lead to better generalisation (and thus lower data prediction errors) with preserved interpretation of fuzzy models. [ABSTRACT FROM AUTHOR]

In this paper we study some fuzzy order in fuzzy sets based on t-norm fuzzy arithmetic. The definition of the order comes from the extension principle for interval order: a > b iff a − b > 0 and from measurement sciences. In measurement sciences the order is given by a comparator whose operation is based on empirical determination of the difference of two input signals. Fuzzy comparison based on fuzzy sets subtraction is considered as an extension of substraction operation, namely a fuzzy set B is greater than a fuzzy set A if B − A is greater than zero in fuzzy arithmetic. In the paper we show that this fuzzy order is irreflexive, transitive, asymmetric and compact, subhomothetic, Archimedean and semi-Ferrers. Our idea refers to the work of M. K. Urbański (Modeling the measurement in algebraic fuzzy structures, Warsaw 2003, in polish) and to the fundamental problems in the measurement theory. [ABSTRACT FROM AUTHOR]

This paper presents a novel hybrid architecture, denoted as EFNN-Nul0 (evolving neuro-fuzzy system based on null-unineurons), meticulously crafted for stress identification within the realm of pattern classification. The model seamlessly integrates neural networks, fuzzy systems, and n-uninorms to cultivate knowledge within an adaptive, real-time learning context. Neuro-fuzzy rules, encapsulating interdependencies among input features through IF-THEN type relationships, are formulated utilizing null-unineurons derived from n-uninorms. This construction enables the articulation of both AND- and OR-connections, thereby augmenting the interpretability of the generated rules. The evolution of neurons is facilitated by an extended adaptation of the autonomous data partition method (ADPA). To dynamically interpret the evolution of rules, the paper introduces (i) a mechanism that tracks changes in rules over data stream samples, providing insights into the process dynamics as a structural active learning component, and (ii) a concept for incrementally updating feature weights. These weights encapsulate the varying impact levels of features on stress identification, enabling the reduction of rule length by effectively masking out less significant features. The outcomes of the rules are represented by certainty vectors, recursively updated through an indicator-based recursive weighted least squares (I-RWLS) approach. Neuron activation levels play a pivotal role in determining the weights, ensuring stable local learning. The effectiveness of the proposed model is substantiated through a comprehensive comparison with existing hybrid and evolving approaches in the literature, focusing on binary and multi-class pattern classification. Aspects that identify special characteristics of neuro-fuzzy models related to interpretability will also be demonstrated in this work. The results demonstrate the model's superior performance, characterized by consistently higher accuracy trend lines over time. Furthermore, the model's interpretability is underscored by the coherent neuro-fuzzy rules, contributing to its remarkable accuracy (the EFNN-Nul0 achieved approximately 99% accuracy in stress identification), establishing its efficacy in addressing stress identification within classification problems. [ABSTRACT FROM AUTHOR]

Fuzzifying the crisp functions is a well-known methodology to study the problems under fuzzy uncertainty. The traditional way for fuzzifying the crisp functions is based on the extension principle. In this paper, by referring to the expression in decomposition theorem, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions obtained from the extension principle and the expression in decomposition theorem under the proposed concept of compatibility. The practical cases regarding the equivalences are also studied in order to be used for the practical problems. [ABSTRACT FROM AUTHOR]

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L -fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces. [ABSTRACT FROM AUTHOR]

Localization Relations are symmetric and T -transitive fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets. This paper shows that not all symmetric and T -transitive fuzzy relations may be obtained in such way, provides an easy condition for characterizing those that may, and states a representation theorem for this class of relations. The logical and geometric implications of such representation are analyzed. The paper also characterizes those (just) symmetric fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive fuzzy tracking control with filter-based event-triggered mechanism for stochastic nonlinear uncertain system suffered input hysteresis is presented. Considering stochastic disturbances and input hysteresis, Fuzzy Logic Systems (FLSs) are adopted to handle such issues. A time-varying vector contained weight vector and approximation error is established. Then, a class of smooth function is introduced to develop the adaptive control laws. It is aiming to avoid the chattering issue. Further, to achieve the desired tracking performance, this paper considers the discontinuous control input caused by the event-triggered control strategy. And the filter-based event-triggered strategy is presented. With the aid of Lyapunov function and Backstepping technique, the fuzzy event-triggered tracking control method is established for the considered system. It is testified that all signals are bounded under the action of the proposed method. Finally, the theoretical results are verified by the presented simulation examples. [ABSTRACT FROM AUTHOR]

This paper studies the adaptive event-based H ∞ control problem for a class of T-S fuzzy distributed parameter system with parameter uncertain and actuator faults. To overcome the drawback of the period control scheme, an event-triggered control scheme with adaptive threshold is proposed to minimize the number of unnecessary sampled data transmission and to reduce the update frequency of the controller. Furthermore, a T-S fuzzy controller based on the adaptive event-triggered sampled data is designed to ensure the stochastic exponential stability of the distributed parameter system with H ∞ disturbance attenuation performance. Based on a new Lyapunov functional and inequality technique, the stochastic exponential stability criterion of the closed-loop system is obtained, and the controller parameters are designed. The new developed inequality can reduce the conservatism of the stability criterion. Finally, the effectiveness of the theoretical calculation results is verified by numerical simulation, and the results are compared with the relevant literature in the simulation, showing that the methods and results in this paper are less conservative. [ABSTRACT FROM AUTHOR]

The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent "TRUE" (increasing functions) and those that represent "FALSE" (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ]) that preserve the function 1. [ABSTRACT FROM AUTHOR]