The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of (S,N)-implications with continuous t-conorms, R-implications obtained from continuous t-norms, Yager's f- and g-generated implications, h- and generalized (h,e)-implications and k-implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation.

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 track 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 appointing these features can accelerate the pace of the future research and be constructive for new researchers.

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions.

The invariance property with respect to powers of continuous t-norms is an additional property of fuzzy implication functions which is particularly interesting in the area of approximate reasoning. In this paper, we provide the characterization of all the binary functions which are invariant with respect to the powers of a nilpotent t-norm and, from the restriction of this result to the definition of fuzzy implication function, we introduce the family of nilpotent T-power invariant implications. We study when the members of this family do satisfy other additional properties apart from the invariance. Moreover, the intersection between the family of nilpotent T-power invariant implications and several families of fuzzy implication functions is investigated.

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this first part we provide three main results: we determine the existence and expression of the additive generator of a cancellative pre-t-norm known above a level curve and, using this result as a basis, we provide the continuous completions of pre-t-norms determined in regions of the type [0,1]2∖(0,a)2 or [0,1]2∖(a,1)2 with a∈(0,1). It is highlighted that, differently from the cancellative case where all the results provide unique continuous completions, in the analogous situations in the conditionally cancellative case the corresponding continuous completions might not be unique. This fact causes more complex results for the remaining regions, which are discussed in the second paper of this study.

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this second part, we use the results of the first part for providing the continuous completions of a conditionally cancellative pre-t-norm defined in the four remaining regions. The most significant contribution in this second part is the study of the case that corresponds to a region of the type B=([0,1]2∖(a,1)2)∪({c}×[0,1])∪([0,1]×{c}) with 0<a<c<1, which has to be separated in six different cases.

This paper aims to compile the main results published in the literature on the study of logical connectives defined on finite chains, as well as to establish relationships between different subfamilies and to set up a common notation for all of them. The operators defined on finite chains which are analyzed are conjunctions and disjunctions such as t-norms, t-conorms, copulas, uninorms, among others; also, results related to implications defined also on finite chains are additionally gathered. For all the theoretical properties, it is mentioned which have been completely or partially solved or, on the contrary, which remain as open problems.

The computation of a closed formula for the cardinality of some discrete connectives has received the interest of the research community since the beginning of this class of operators. This paper constitutes a substantial progress in this topic. First, monotonicities and other properties of discrete connectives are related to plane partitions, a concept deeply studied in the field of combinatorics. Second, the already known expressions on the cardinality of plane partitions are adapted to the concrete properties of discrete connectives. With this, we establish closed formulas for discrete negations and some binary discrete connectives; concretely, discrete conjunctions, disjunctions and implications. As well, we establish formulas for the cardinality of some sets of implications satisfying several additional properties known in the literature.

Determining the number of discrete operators has been a topic of interest for the scientific community since the introduction of these operators. This paper represents a further stage within this topic in the field of operators defined on a finite chain. Mainly, two families of discrete operators are studied: discrete conjunctions that are smooth (a property that is usually considered the equivalent to continuity in discrete settings) and commutative discrete conjunctions with n, the greatest element of the chain, as neutral element, so that only associativity is missing to become discrete t-norms. To determine the cardinality of the first family, we study its explicit representation by alternating sign matrices, obtaining that the cardinality is preserved between both structures and allowing us to relate intrinsic properties of the family of discrete operators with intrinsic properties of such class of matrices. For commutative discrete conjunctions with n as neutral element, we have considered the concepts of n-Gog y n-Magog triangles, allowing us to transform properties of these operators into properties of these triangular arrays; in particular, their cardinality. In this way, an upper bound for the cardinality of discrete t-norms is achieved.

Discrete implications have been studied for almost two decades as those operators needed to perform inference processes when dealing with qualitative information from a finite chain. However, it is known that, by means of some adequate transformations, fuzzy logic operators defined in [0,1] can generate the corresponding discrete operators. Thus, an immediate question arises: Do we need to study discrete implications or is it enough to study implications defined in [0,1], and then, to discretize them? The answer must rely on the preservation of the additional properties of fuzzy implication functions through these discretization methods. In this article, for two specific methods based on the ceiling and floor functions, it is proved that most of the additional properties are not preserved in general, showing that the preservation of the additional properties depends directly on the properties of the underlying operators considered in the discretization. Thus, sufficient, and for some properties necessary, conditions to guarantee the preservation are presented.

From the beginnings of fuzzy logic, the Sheffer stroke operation has been overlooked and the efforts of the researchers have been devoted to other logical connectives. In this paper, the Sheffer stroke operation is introduced in fuzzy logic generalizing the classical operation when the truth values are restricted to {0,1}2. Similar to what happens in Boolean logic, the fuzzy Sheffer stroke is functionally complete and it can be used to generate any other fuzzy logical connective by combinations of itself. Two construction methods are presented and the close connection of this operation with a pair of fuzzy conjunction and negation is analysed.

Characterizations of families of fuzzy implication functions are a necessary step to fully understand the behaviour of the members of the family, their potential applicability and their relations with other families. These characterizations are based on additional algebraical properties that completely define the family. Recently, the class of power based implications was characterized through, among others, the property I ( x , y ) ⋅ I ( y , z ) = I ( x , z ) in a concrete sub-domain. This property, called in the literature also as multiplicative Sincov's equation, was uncommon to other families, and it was studied in-depth in a previous article. This equality is generalized in this paper by understanding the internal product as the product t-norm and changing it to a general arbitrary continuous Archimedean t-norm. This additional property is analyzed jointly with the weak ordering property or the ordering property, leading to a characterization of those fuzzy implication functions satisfying both additional properties.

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.

In the last decades, more than a hundred families of fuzzy implication functions have been proposed. These families are generated by adequately combining other logical connectives or univariate functions. However, no attention has been given to the final expression of the fuzzy implication function, a crucial feature for any application. In this paper, fuzzy polynomial implications are introduced as those fuzzy implication functions whose expression is given by a polynomial of two variables. Polynomials present advantages with respect to other types of functions making them interesting for applications. Several results are proved for polynomials of any degree and the characterisation of all fuzzy polynomial implications of degree less or equal to 4 is achieved.

The linguistic computational model based on discrete fuzzy numbers has gained great interest among scholars due to its interesting properties. However, the investigations on group consensus with this linguistic model are not enough and need further exploration. For this reason, in this paper, we propose a novel consensus model based on this framework that overcomes some of the main disadvantages of the previously proposed methods in the literature. Moreover, we provide a new aggregation function on the set of discrete fuzzy numbers and we propose a semi-automatic algorithm that allows the experts to interact and modify their opinions during the consensus process. This novel method achieves a significantly greater rate of convergence in Group Decision Making problems compared with the existing algorithms.

The so-called family of T-power based implications has been introduced recently by using Zadeh's quantifiers modelled by powers of t-norms in its definition. Most of these operators satisfy the invariance with respect to powers of a continuous t-norm, an important property in approximate reasoning. When this family of fuzzy implication functions was characterized, the property I(x, y) · I(y, z) = I(x, z) in a concrete sub-domain played a key role. This property, which ensures that T-power based implications are unidimensional T'-preorders with T' the product t-norm, seems to be related to the invariance property. Therefore, the natural question of characterizing some classes of fuzzy implications with this property arises naturally. We provide such a characterization in a fairly general setting, generalizing earlier known results.

Among the great bunch of additional properties which fuzzy implication functions may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication function when both the antecedent and the consequent are modified using the same quantifier modelled using powers of t-norms. In this paper, a related property called T-power inverse invariance property is introduced. This property ensures the invariance of the truth value of the fuzzy implication function when the consequent is modified using the quantifier inverse to the one used to modify the antecedent. The characterization of those binary mappings I : [0, 1]2→ [0, 1] fulfilling this property for continuous Archimedean t-norms is presented as well as the particular characterization result for fuzzy implication functions.

The law of importation is an important property of fuzzy implication functions with interesting applications in approximate reasoning and image processing. This property has been extensively studied and some open problems have been proposed in the literature. In particular, in this paper, we partially solve an open problem related to this property posed some years ago. Specifically, given a fixed uninorm, all fuzzy implication functions that satisfy the law of importation with respect to this uninorm, and having an α-section that is a continuous negation, are characterized. This characterization is specially detailed for the case of uninorms lying in each one of the most usual classes of uninorms. This is done in two different papers, this paper and the forthcoming paper (S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with a fixed uninorm-Part II,”IEEE Trans. Fuzzy Syst., to be published). In particular, in this paper the case of uninorms inUminis solved, whereas the cases where the uninorm is in the other usual classes (that is, idempotent, representable, and continuous in the open unit square) are left for the above-mentioned paper.

The law of importation is an important property of fuzzy implication functions because of its interesting applications in approximate reasoning and image processing. In this paper, as a continuation of the paper [S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with respect to a fixed uninorm-Part I,” IEEE Trans. Fuzz Syst., to be published.], the characterization of all fuzzy implication functions that satisfy the law of importation with respect to a given uninorm U, and having an α-section that is a continuous fuzzy negation is given for the cases when the uninorm U lies in one of the most used classes of uninorms. As the case when the uninorm U is inUmin was already solved in [S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with respect to a fixed uninorm-Part I,” IEEE Trans. Fuzz Syst., to be published.], this paper focuses in the cases when U is an idempotent uninorm, a representable uninorm, or a uninorm continuous in the open unit square.

Morphological operators have been used extensively when dealing with binary or grayscale images, but there is no general-purpose approach for multivariate images that meets the expectactions of practitioners under diferent circumstances. Although several approaches have been proposed, state-of-the-art applications tend to process channels independently, obviating interchannel correlation. In this work, we introduce a new definition of erosion and dilation that can handle images with any number of channels, study their theoretical properties and analyse its behaviour. It is based on the Fuzzy Mathematical Morphology, from which it inherits essential theoretical properties. Our operators consider the first channel to evaluate a pixel's importance, but handles all channels to generate coherent outputs. It successfully processes natural images in the L*a*b* space and can also avoid the creation of new chromatic values, specially important for hyperspectral imagery. We provide, thus, a general and well-founded framework to process color images with morphological operators.

In many computer vision applications, vignetting is an undesirable effect which must be removed in a pre-processing step. Recently, an algorithm for image vignetting correction has been presented by means of a minimization of log-intensity entropy. This method relies on an increase of the entropy of the image when it is affected with vignetting. In this paper, we propose a novel algorithm to reduce image vignetting via a maximization of the fuzzy entropy of the image. Fuzzy entropy quantifies the fuzziness degree of a fuzzy set and its value is also modified by the presence of vignetting. The experimental results show that this novel algorithm outperforms in most cases the algorithm based on the minimization of log-intensity entropy both from the qualitative and the quantitative point of view.

Copulas are a special kind of aggregation functions that have been deeply investigated because of their applications in many fields, specially in Statistics and Economy. An important research topic from the theoretical point of view is the study of new construction methods of copulas. In this line, this paper presents two construction methods based on probabilistic implications and survival implications. From these construction methods, the axiomatic characterization of these families of fuzzy implication functions, which are in fact the same, is presented.

In this paper aggregation functions whose expressions are given by polynomial functions are investigated. A detailed study focused on binary polynomial aggregation functions of degree one and two is given not only in general, but also requiring some additional properties like idempotency, commutativity, associativity, one-side neutral (or absorbing) element and so on, leading to some families of binary polynomial aggregation functions. The results concerning polynomials of degree one are generalized to n-ary polynomial aggregation functions whereas, with respect to those of degree two, the commutative case is also generalized obtaining the characterization of all commutative n-ary polynomial aggregation functions of degree two in general and jointly with the idempotent property.

The law of importation is an important property of fuzzy implication functions with interesting applications in approximate reasoning and image processing. This property has been extensively studied, and some open problems have been posed in the literature. In particular, in this paper, we partially solve an open problem related to this property posed some years ago. Specifically, given a fixed t-norm T , all fuzzy implication functions with continuous natural negation that satisfy the law of importation with this t-norm T are characterized. This characterization is especially detailed for the case of any continuous t-norm T , and particular cases are given for the minimum t-norm, for any continuous Archimedean t-norm, and for any ordinal sum of continuous Archimedean t-norms. For noncontinuous t-norms, the particular cases of the drastic t-norm and the nilpotent minimum t-norm are also presented separately. Finally, characterizations of some well-known fuzzy implication functions are also deduced from the presented results.

The proposal of new edge detectors is a constant in last years due to the importance of the edge detection result for high-level image processing tasks. It is known that there is no optimal edge detector for all kinds of images and in addition, it is not a straightforward mission to set the optimal set of parameters of the edge detector for each image. In this paper, an algorithm to deal with these drawbacks is presented based on obtaining a consensus edge image from the edge images obtained by several edge detectors or configurations of the same edge detector. These input edge images are combined using multifuzzy sets and averaging aggregation functions and after applying a penalty function, the consensus edge image is obtained. Experimental results show that this approach leads to a robust edge detector which is able to obtain notable results for different kinds of images.

The proposal of new classes of fuzzy implication functions must take into account the final expression of the operator to be potentially used in a concrete application. In particular, fuzzy implication functions with simple expressions have low computational cost and they reduce the spreading of numerical errors. Following this line of research, in this work, the class of fuzzy rational implications is introduced as those fuzzy implication functions whose expression is given by the quotient of two polynomials of two variables. Since not all quotients of polynomials are adequate to generate a fuzzy implication function, some necessary properties related to the values of the coefficients of the polynomials are given to ensure obtaining a fuzzy implication function. In addition, we characterize all fuzzy rational implications constructed from polynomials of some fixed degrees. Finally, several construction methods of these implications and relationships with other families of fuzzy implication functions are given.

Noise reduction is a fundamental step in many image processing and computer vision applications. In recent years, several noise filters especially devoted to the removal of high density salt and pepper noise, as a particular case of impulse noise, have been proposed. In this paper, a novel two-stage filter based on the fuzzy mathematical morphology using t-norms and aggregation functions is proposed. This filter involves a detection step of the noisy pixels and the restoration of the image by means of the aggregation of the non-corrupted pixels through an adequate weighted arithmetic mean. The experimental results show that the proposed algorithm outperforms other filtering methods both from the visual point of view and the values of some objective performance measures for images corrupted from a 25% up to 98% of noise.

In this paper, a novel consensus model for group decision making based on discrete fuzzy numbers is proposed. The experts express their preferences over the alternatives by means of the so-called subjective linguistic preference relations, which allow a greater flexibility of the experts' opinions. The model consisting of a consensus and a selection phases is able to achieve the consensus without any loss of information and without imposing to the experts any drastic change on their initial opinions. Finally, a group decision making problem based on this model is presented.

In this paper, the fuzzy morphological gradients from the fuzzy mathematical morphologies based on t-norms and conjunctive uninorms are deeply analyzed in order to establish which pair of conjunction and fuzzy implications are optimal, in accordance with their performance in edge detection applications. A novel three-step algorithm based on the fuzzy morphology is proposed. The comparison is performed by means of the so-called Pratt's figure of merit. In addition, a statistical analysis is carried out to study the relationship between the different configurations and to establish a classification of the conjunctions and implications considered. Both the objective measure and the statistical analysis conclude that the pairs nilpotent minimum t-norm and the Kleene-Dienes implication, and the idempotent uninorm obtained with the classical negation as a generator and its residual implication, are the best configurations in this approach, because they also obtain competitive results with respect to other approaches.

Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.

In this paper, a new approach to fuzzy mathematical morphology based on discrete t-norms is studied. It is proved that the most usual algebraic and morphological properties are preserved, such as, duality, monotonicity, interaction with union and intersection, invariance under translating and scaling, local knowledge property, extensitivity, idempotence, and many others. In fact, all properties satisfied by the approach based on nilpotent t-norms hold in the discrete case. This is quite important since in practice we only work with discrete objects. Experimental results for some discrete t-norms are included. They are compared with classical morphological algorithms based on the Łukasiewicz t-norm and the umbra approach, and with the fuzzy approach based on idempotent uninorms, proving that they are suitable to be used in edge detection.