The law of contraposition is one of the most well-known tautologies in classical logic. In addition, there are two other notions of contrapositions: the law of right contraposition and the law of left contraposition. In fuzzy logic, these tautologies are modeled by a fuzzy implication and a fuzzy negation. The characterization of fuzzy implications satisfying the laws of right or left contraposition with respect to an arbitrary fuzzy negation is a problem that has gained attention recently. This article aims to propose characterizations of such implications, presenting necessary and sufficient conditions under which a fuzzy implication satisfies these laws with respect to an arbitrary fuzzy negation.

The topic of fuzzy and relation-changing structures has been progressively explored through the authors' previous work. This paper continues to expand the range of application for these models by introducing Labeled Fuzzy Reactive Graphs. This structure admits labels in regular edges, in order to permit to describe the action of different agents. For such structures, the operations of union, intersection and threshold are presented. Subsequently, this structure is applied to model a cancer treatment protocol and it is shown how the relation-changing properties of this structure can be used to better describe the dynamics of the modeled protocol.

Ordered Weighted Averaging (OWA) operators are some of the most widely used aggregation functions in classic literature, but their application to fuzzy numbers has been limited due to the complexity of defining a total order in fuzzy contexts. However, the recent notion of admissible order for fuzzy numbers provides an effective method to totally order them by refining a given partial order. Therefore, this paper is devoted to defining OWA operators for fuzzy numbers with respect to admissible orders and investigating their properties. Firstly, we define the OWA operators associated with such admissible orders and then we show their main properties. Afterward, an example is presented to illustrate the applicability of these AOWA operators in linguistic decision-making. In this regard, we also develop an admissible order for trapezoidal fuzzy numbers that can be efficiently applied in practice.

The idea of nondeterministic typical hesitant fuzzy automata is a generalization of the fuzzy automata presented by Costa and Bedregal. This paper, presents the sufficient and necessary conditions for a typical hesitant fuzzy language to be computed by nondeterministic typical hesitant fuzzy automata. Besides, the paper introduces a new class of typical hesitant fuzzy automata with crisp transitions, and we will show that this new class is equivalent to the original class introduced by Costa and Bedregal.

Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is calledoverlap indices. They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion ofoverlap indicesis also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion oftype-$(2,k)$overlap index($k \in \lbrace 0,1,2\rbrace$) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type-$(2,k)$overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.

A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor.This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism.However, the effect of parameter λ remains unchanged for all the input values.In this study, we generalize further the condition of abstract homogeneity by introducing ℱ-homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs.Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions.To achieve that, we generalize directional monotonicity by ℱ directional monotonicity, which is defined with respect to a family of functions ℱ and a family of vectors  .Finally, we show how the introduced concepts could be applied

Traditionally, Convolutional Neural Networks make use of the maximum or arithmetic mean in order to reduce the features extracted by convolutional layers in a downsampling process known as pooling. However, there is no strong argument to settle upon one of the two functions and, in practice, this selection turns to be problem dependent. Further, both of these options ignore possible dependencies among the data. We believe that a combination of both of these functions, as well as of additional ones which may retain different information, can benefit the feature extraction process. In this work, we replace traditional pooling by several alternative functions. In particular, we consider linear combinations of order statistics and generalizations of the Sugeno integral, extending the latter's domain to the whole real line and setting the theoretical base for their application. We present an alternative pooling layer based on this strategy which we name "CombPool" layer. We replace the pooling layers of three different architectures of increasing complexity by CombPool layers, and empirically prove over multiple datasets that linear combinations outperform traditional pooling functions in most cases. Further, combinations with either the Sugeno integral or one of its generalizations usually yield the best results, proving a strong candidate to apply in most architectures.

Throughout this article, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first, we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice—where well-known impossibility results as the Arrovian ones are encountered—and the decision-making approaches—where the necessity of fusing rankings is unavoidable. Moreover, it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then, we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice, one should look for the maximal elements with respect to such orders defined on rankings.

In this work, we generalize the notion of restricted equivalence function for type-2 fuzzy sets, leading to the notion of extended restricted equivalence functions. We also study how under suitable conditions, these new functions recover the standard axioms for restricted equivalence functions in the real setting. Extended restricted equivalence functions allow us to compare any two general type-2 fuzzy sets and to generate a similarity measure for type-2 fuzzy sets. The result of this similarity is a fuzzy set on the same referential set (i.e., domain) as the considered type-2 fuzzy set. The latter is crucial for applications such as explainable AI and decision-making, as it enables an intuitive interpretation of the similarity within the domain-specific context of the fuzzy sets. We show how this measure can be used to compare type-2 fuzzy sets with different membership functions in such a way that the uncertainty linked to type-2 fuzzy sets is not lost. This is achieved by generating a fuzzy set rather than a single numerical value. Furthermore, we also show how to obtain a numerical value for discrete referential sets.

In this article, we propose a new generalization for the notion of homogeneous functions. We show some properties and how it appears in some scenarios. Finally, we show how this generalization can be used in order to provide a new paradigm for decision-making theory calledconsistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.

Generalized Mixture (GM) and Bounded Generalized Mixture (BGM) functions, as well as OWA and Choquet integrals, are weighted averaging functions which provide a broad class of fusion functions. However, both, OWA and Choquet Integrals, satisfy the monotonicity condition, which fails for some GM and BGM functions. In this paper we investigate two generalized forms of monotonicity: The directional monotonicity and ordered directional monotonicity, then we determine some criteria for obtaining GM and BGM functions which satisfy such properties.

In this paper we present a subvariety of the naBL-algebras based on overlaps functions, as well as a version of the well-known Chinese Remainder Theorem for naBL-algebras. Moreover notions of automorphism and its action on overlaps is used to obtain naBL-conjugated algebras.

The use of fuzzy implications is widespread as easily found in the literature. The study of fuzzy implications attracts the attention of many authors probably because of their theoretical features and also the diversity of applications where they can be applied. In the present work we continue the study of a type of fuzzy implication, called (T, N)-implication, constructed by joining a fuzzy negation and a t-norm. However, we generalize those (T, N)-implications by presenting a new operator, named (N', T, N)-implication, which can be derived from two negations. We also study the properties of (N', T, N)-implications and compare the new results with the previous ones.

Usually, the evaluation of the classifiers performance is not an easy task to be performed, mainly when we analyze different criteria (output parameters). In this evaluation process, we can use quantitative measures (accuracy, specificity, among others), however, when the output values are very close and we have several criteria, the results are difficult to be interpreted by users. This paper aims to propose a new linguistic model to evaluate the performance of several classifiers. It is based on the notion of gradual complex numbers (GCN), proposed in [18]. In this paper, we present the theoretical basis of GCNs for classifier evaluator and we assess the performance of the proposed model (GCN) through an empirical study. In addition, the performance of GCN is compared with that of fuzzy complex numbers[6], and it reveals gains.

The aim of this work is to study a certain fuzzy implication, called (T, N)-implication, obtained by composition of a fuzzy negation and a t-norm. Thus, it discusses under which conditions such functions preserve the main properties of fuzzy implications, in which some are related to the laws of contraposition. Finally, we prove a result that ensure the necessary and sufficient conditions for a function I : [0,1]2→ [0,1] to be a (T, N)-implication.

Generalized Mixture (GM) functions beyond to generalize the Mixture functions, also generalizes an important type of aggregation functions, called Ordered Weighted Averaging (OWA). An OWA is a parametrized average, it uses previously defined weights. A GM is also parametrized average, but with a not fixed predefined vector of weights. Whereas applications of OWA depends on the determination of the best vector of weights to be applied, the vector of weights for GM's is determined as a function of input arguments. In this paper we present the notion of GM, show some of their proprieties and build a way to create GM functions starting from any other function of the form: Θ ∶ [0, 1]n→ [0, 1].

The main goal of using data with interval nature is to represent numeric information endowed with impreciseness, which are normally captured from measures of real world. However, in order to do this, it is necessary to adapt real-valued techniques to be applied on interval-based data. For interval-based clustering applications, for instance, it is necessary to propose an interval-based distance and also to adapt clustering algorithms to be used in this context. Therefore, in this paper, we aim to provide a platform for performing clustering applications using interval-based data, including distance measure, clustering algorithms, and validation indexes. In this case, we adapt an interval-based distance called dkm, and we propose two interval-based fuzzy clustering algorithms: Interval-based FcM and interval-based ckMeans, and three interval-based validation indexes. In order to validate the proposed interval-based framework, an empirical analysis was conducted using seven clustering datasets, three real and four synthetic interval datasets. The empirical analysis is based on an external cluster validity index, corrected rand, and six internal-based validation indexes, in which three of them can be used in their original proposal and three are proposed in this paper. The obtained results show the usefulness of the proposed interval-based framework for interval-based clustering problems.

In this paper we define the concept of uniformly strongly prime fuzzy ideal for associative rings with unity. This concept is proposed without dependence of level cuts. We show a pure fuzzy demonstration that all uniformly strongly prime fuzzy ideals are a prime fuzzy ideal according to the newest definition given by Navarro, Cortadellas and Lobillo [1] in 2012. Also, some properties about fuzzy strongly prime radical and their relations with Zadeh's extension are shown.

In this paper we introduced the concept of three-dimensional triangular fuzzy number and their properties are investigated. It is shown that this set has important metrical properties, e.g convexity. The paper also provides a rotation method for such numbers based on quaternion and aggregation operator.

The main advantage of using an interval-based distance for interval-based data lies on the fact that it preserves the underlying imprecision on intervals which is usually lost when real-valued distances are applied. One of the main problems when using interval-based distance in fuzzy clustering algorithms is the way to obtain the center of the groups. In this case, it is necessary to make adaptations in order to obtain those centers. Therefore, in this paper, we propose the use of the family of H-operator to proposed three approaches to transform the interval-based membership matrix into real-valued membership matrix and, as a consequence, to calculate the centers of the groups in interval-based fuzzy clustering algorithms. In this case, we will perform a comparative analysis using the three different approaches proposed in this paper, using seven interval-based datasets (four synthetic and three real datasets). As a result of this analysis, we will observe that the proposed approaches achieved better performance than all analyzed methods for interval-based methods.

Since the seminal paper of fuzzy set theory by Zadeh in 1965, many extensions have been proposed to overcome the difficulty for assigning the membership degrees. In recent years, a new extension, the hesitant fuzzy sets, has attracted a lot of interest due to its usefulness to handle those problems in which it is difficult to provide accurately a single membership value; since for hesitant sets, membership values are given by a whole set of values. On the other hand, since fuzzy negations have an important role in applications as well as in the theoretical approach to of fuzzy logics, it is important to study an extension of the concept of fuzzy negation for hesitant fuzzy degrees (elements). In this paper, we propose such a definition and we study some of the main properties of this new concept.

The well-known Boolean-law “α → (β → α) = 1” can be generalized to fuzzy context as I(x, I(y, x)) = 1, where I is a fuzzy implication. In this paper we show the necessary and sufficient conditions under which this generalization holds in fuzzy logics. We focus the investigation on the following classes of fuzzy implication: (S,N)-, R-, QL-, D- and (N, T)-implications. In addition, we demonstrate that a fuzzy implication I satisfies such Boolean-like law if, and only if, its Ф-conjugate also satisfies it.

In this paper it is defined the concept of strongly prime fuzzy ideal for noncommutative rings. Also, it is proved that the Zadeh's extension preserves strongly fuzzy primeness and that every strongly prime fuzzy ideal is a prime fuzzy ideal as well as every fuzzy maximal is a strongly prime fuzzy ideal.

In this paper we build the concept of fuzzy quaternion numbers as a natural extension of fuzzy real numbers. We discuss some important concepts such as their arithmetic properties, distance, supremum, infimum and limit of sequences.

Properties valid on classical theory (Boolean laws) have been extended to fuzzy set theory. Such generalizations of Boolean laws (Boolean-like laws) are not always valid in any standard fuzzy set theory and this fact induced a wide investigation. In this paper we show the conditions that the Boolean-like law x ≤ I(y; x) holds in fuzzy logic. We focus the investigation on three main classes of fuzzy implications, namely: (S,N)-, R- and QL-implications. Further, we show that the operator I satisfies this Boolean-like law if, and only if, Φ-conjugate of I also satisfies it.

We characterize a fuzzy lattice through a fuzzy partial order relation, define a fuzzy ideal and fuzzy filter of fuzzy lattice, characterize a fuzzy ideal of fuzzy lattice using its level set and its support and show that a subset of a fuzzy lattice is a fuzzy ideal if and only if its support is a crisp ideal. Similarly, we show the same for its level set.

In this paper, we propose a natural way to define topologies by using fuzzy metrics; here the value of the distance between two points is a non-negative fuzzy number. With such topology, we also define notions like continuity, convergent sequence, Cauchy sequences and complete spaces. As an application, we provide a fixed point theorem quite similar to the classical Banach's theorem.

In this article, we use the interval mathematics and targeted rounding by specific functions to establish a framework for interval quantization. The function approximation F Id , that maps x to an interval [x 1 , x 2 ] such that x 1 is the largest floating point number less than or equal to x and x 2 is the smallest floating point number greater than or equal to x, is used to establish the sampling interval and the levels of interval quantization. We show that the interval quantization levels (N j ) represent the specific quantization levels (n j ), that are comparable, according to Kulisch- Miranker order and are disjoint two by two. If an interval signal X[n] intercepts a quantization interval level N j , then the quantized signal will be X q [n] = N j . Moreover, for the interval quantization error (E[n] = X q [n] − X[n]) an estimate is shown due to the quantization step and the number of levels. It is also presented the definition of interval coding, in which the number of required bits depends on the amount of quantization levels. Finally, in an example can be seen that the the interval quantization level represent the classical quantization levels and the interval error represents the classical quantization error.

The aim of this work is to analyze interval fuzzy S-implications and interval automorphisms. Starting from any fuzzy S-implication, it is shown how to obtain an interval fuzzy S-implication canonically. We proved that such interval fuzzy S-implications meet the optimality property and preserve the same properties satisfied by fuzzy S-implications. In addition, commutative diagrams are used in order to relate fuzzy S-implications to interval fuzzy S-implications, and to understand how interval automorphisms act on interval S-implications, generating other interval fuzzy S-implications.

Abstract An interval is a continuum of real numbers, defined by its end-points. Interval analysis, proposed by R. Moore in the 50's, concerns the discovery of interval functions to produce bounds on the accuracy of numerical results that are guaranteed to be sharp and correct. The last criterion, correctness , is the main one since it establishes that the result of an interval computation must always contains the value of the related real function. Correctness rests on the ``Fundamental Theorem of Interval Arithmetic''. This theorem, induced us to define interval representations , which captures the fact that interval analysis works like a kind of language to express computations with real numbers, this is implicitly formulated at [HJE01] p. 1045, lemma 2. Until now the idea of intervals as representation of real numbers was not explicitly defined and its relation with some aspects of interval analysis was not explored. Here we show some of these relations in terms of the topological aspects of intervals (Scott and Moore topologies). The paper also defines what we call the canonical interval representation of a real function, which is the obvious best ``mathematical'' (not necessarily computable) interval representation of a real function [inline-graphic not available: see fulltext]. The idea is to show some properties of correct interval algorithms giving a relationship with some other kind of functions, such as extensions and inclusion monotonic functions; and to show how this ideal object preserves the continuity of real functions into Scott and Moore topologies.

The most normal forms for fuzzy logics are versions of conjunctive and disjunctive classical normal forms. Unfortunately, they do not preserve neither 1-tautologies nor 0-contradictions. This paper introduces a normal form that partially preserves 1-tautologies for any continuous t-norm - i.e. if a formula is a 1-tautology then their normal form is also a 1-tautology but the reciprocal does not always hold. For the class of t-norms without zero divisors it preserves 0-contradictions, i.e. a formula is 0-contradiction if and only if their normal form is also 0-contradiction. The paper shows that this normal form could be used to implement an automatic theorem provers for a class of residuum-based propositional fuzzy logics.