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Abstract: In this paper residual implications defined on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers are studied. A specific construction of these implications is given and some examples are presented showing in particular that such a construction generalizes the case of interval-valued residual implications. The most usual properties for these operations are investigated leading to a residuated lattice structure on the set of discrete fuzzy numbers, that in general is not an MTL-algebra. [Copyright &y& Elsevier]

Abstract: In this paper, characterizations of Yager’s f- and g-implications are presented. Since their introduction in 2004, the properties of these implications have been studied in detail but they have not been characterized yet. The characterizations given here are based on the law of importation, a functional equation that has been extensively studied. Moreover, particular characterizations of Reichenbach, Yager and Goguen implications are derived from these results. [Copyright &y& Elsevier]

Abstract: The intersection of the different classes of implications is one of the most popular topics nowadays due to the large number of construction methods of these operators. In this paper, we deal with the characterization of the intersection of Yager’s implications with QL and D-implications. Some initial steps have already been made with the intersection of Yager’s implications with (S, N), R and QL-implications, however some questions remain unanswered. In particular, we solve an open problem related to the characterization of those implications that are both QL-implications and f-generated implications with f(0)<∞, fully determining the expression of the QL-implications generated by a continuous t-conorm that belong to the considered intersection. Furthermore, we perform a similar study for D-implications and finally, we study the intersection of Yager’s implications with their φ-conjugates. [Copyright &y& Elsevier]

Abstract: A new class of fuzzy implications called the h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yager’s f- and g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called (h, e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function F : [0,1]2 →[0,1]. [Copyright &y& Elsevier]

Abstract In Massanet et al. (2017) [1] a new property of fuzzy implication functions, called the invariance property with respect to powers of a continuous t-norm, was introduced and its application in approximate reasoning was studied. In the same paper, the novel family of power based implications was presented as a family of fuzzy implication functions that satisfy such invariance property. Unfortunately, this fact is not entirely true since as we will prove in this paper, there exist some power based implications generated from specific ordinal sum t-norms which do not fulfil the invariance property. Thus, a characterization of which continuous t-norms can be used to generate power based implications satisfying the power based invariance property is presented. Additionally, as an alternative solution, we introduce in this paper a slight modification of this property in such a way that all power based implications satisfy it. [ABSTRACT FROM AUTHOR]

The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions. [ABSTRACT FROM AUTHOR]

In a group decision making problem, experts often express their opinions through the so-called preference relations. In recent years, several different definitions of preference relations depending on the framework and the nature of the problem have been introduced. These approaches vary from interval-valued fuzzy preference relations to incomplete fuzzy linguistic preference relations. In this paper, a novel definition of preference relation, the so-called subjective linguistic preference relation, is proposed. These preference relations are based on the concept of subjective evaluations, introduced in the linguistic computational model based on discrete fuzzy numbers. In this framework, the experts have more flexibility to express their opinions and the solid mathematical background of this model is a guarantee of no loss of information. Finally, an example of a multi-expert decision making problem with a hierarchical multi-granular linguistic context is analyzed to illustrate the potential of the proposed method and its advantages with respect to other methods. [ABSTRACT FROM AUTHOR]

Abstract: In recent years, several different linguistic computational models for dealing with linguistic information in processes of computing with words have been proposed. However, until now all of them rely on the special semantics of the linguistic terms, usually fuzzy numbers in the unit interval, and the linguistic aggregation operators are based on aggregation operators in [0,1]. In this paper, a linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is presented ensuring the accuracy and consistency of the model. In this framework, no underlying membership functions are needed and several aggregation operators defined on the set of all discrete fuzzy numbers are presented. These aggregation operators are constructed from aggregation operators defined on a finite chain in accordance with the granularity of the linguistic term set. Finally, an example of a multi-expert decision-making problem in a hierarchical multi-granular linguistic context is given to illustrate the applicability of the proposed method and its advantages. [Copyright &y& Elsevier]

In fuzzy logic and approximate reasoning the inference rule given by the Modus Tollens usually derives into an inequality involving three logical operators: a conjunction, an implication function and a negation. Until now, in this scenario the conjunction has been commonly modeled by a t-norm, but recently the possibility of using a more general conjunction has been pointed out. In this work, we want to generalize the Modus Tollens inequality by using a conjunctive uninorm instead of a t-norm, leading to the so-called U -Modus Tollens. First, we give a study of this new property for implication functions in general and then we specially focus on residual implications derived from uninorms. In all cases, we prove that there are a lot of solutions of the U -Modus Tollens and we give a characterization of all the solutions in some particular cases. [ABSTRACT FROM AUTHOR]

In the last decade, several new construction methods of fuzzy implication functions from one or more given ones have been proposed. Following this line of research, in our paper some construction methods based on polynomial functions of three variables are presented. Concretely, these methods provide a (possibly new) fuzzy implication function from a given one and a quadratic polynomial function. A complete characterization of those quadratic functions adequate to obtain a fuzzy implication function through this strategy is presented. Moreover, the invariant fuzzy implication functions with respect to this method are determined. Finally, some quadratic construction methods preserving some specific additional properties are analysed. [ABSTRACT FROM AUTHOR]

Abstract: Fuzzy operations acting on entire fuzzy sets with the stress on fuzzy implications are discussed and studied. In the case of binary operations, the input fuzzy sets are fuzzy subsets of possibly different universal spaces X and Y, and the output fuzzy set is a fuzzy subset of the Cartesian product . The standard approach to fuzzy operations is based on functions acting on , and then these fuzzy operations are called functionally expressible. We give a characterization of functionally expressible fuzzy implications (and other fuzzy operations), and include several examples of fuzzy operations which are not functionally expressible. [Copyright &y& Elsevier]