Lianget al.in 2000 defined interval type-2 fuzzy sets (IT2FSs), which constitute a subset of type-2 fuzzy sets. While the membership degrees in the former are functions from [0, 1] to [0, 1] (fuzzy truth values), the membership degrees in IT2FSs only take their values in$\lbrace {\text{0}},{\text{1}}\rbrace$. Although all the initial work on IT2FSs involved convex membership degrees only, in 2015, Bustinceet al.began the study on IT2FSs in general, including certain sets with nonconvex membership degrees. However, these are obviously early stages, with a lot of open problems regarding the theoretical structure of IT2FSs. For example, as far as we know, no negation operator has been obtained in this context. Therefore, it seems appropriate to continue with the study started in previous papers, delving deeper into the properties and operations of IT2FSs. Consequently, this work studies the structure of the set of functions from [0, 1] to$\lbrace {\text{0}},{\text{1}}\rbrace$(expanding the set considered by Bustinceet al.), from which we have removed the constant function$\mathbf{0}$, to offer a different study to the one carried out by Walker and Walker. More specifically, we consider join and meet operations, partial order derived from each one, and the negation operators in that set. Among other results, we provide new characterizations of join and meet operations and of partial orders on the set of functions from [0, 1] to$ \lbrace {\text{0}},{\text{1}} \rbrace $; we also present the first negation operators on this set.

The concept of pre-aggregation function defined in [0,1]nhas been recently extended to that of generalized pre-aggregation function in the framework of a totally ordered set T with maximum and minimum value. To do so, the concept of monotonicity is transformed in that of conditioned monotonicity based on the chains in Tn, generalizing the idea of directional monotonicity. In the present paper we explore the concept of conditioned monotonicity considering some specific conditioning structures (covers, partitions and projections). On this basis we consider some situations where conditioned monotonicity ensures monotonicity. Finally we use these definitions and properties to define some pre-aggregation and aggregation functions that are applied to image preprocessing problems.

Type-2 fuzzy sets (T2FSs) were introduced by L. A. Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). In this extension, the degree to which an element belongs to a set is just a label of the linguistic variable “TRUTH,” which allows to represent reality in a more appropriate way. On the other hand, negations play an essential role within fuzzy sets theory. In fact, they are necessary in order to obtain, for example, complements of fuzzy sets, dual of at-norm or a t-conorm, entropies, implications, as well as to study the possible contradictions appearing in a fuzzy system. However, meanwhile negations on [0, 1] (set of the membership degrees of a fuzzy set) have been deeply studied throughout the literature, the same has not happened with the negations on M = Map ([0, 1], [0, 1]), set of functions from [0, 1] to [0, 1] (and also set of the membership degrees of a T2FS), and so many aspects of the negations on M have not yet been investigated. In a previous paper, the axioms that an operation must satisfy to be considered a negation or a strong negation on a bounded partially ordered set were established. A family of strong negations on L and set of normal and convex functions of M were also presented. Moreover, let us note that the main characteristic of fuzzy systems is just the flexibility in order to be able to represent knowledge according to each situation, offering different models among which the expert can choose the one that best suits his/her criteria. Thus, it seems useful to find broad sets of negations in M and in L, which, as far as we know, have not been done by other researchers. According to these ideas, in this paper, the authors first present new negations and strong negations on L, and then show, for the first time, some negations on M with respect to each of the two partial orders defined in this set.

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets. The degree of membership of an element for T2FSs is a fuzzy set in [0, 1], that is, a T2FS is determined by a membership function from the universe of discourse X to M, where M is the set of functions from [0, 1] to [0, 1]. Walker and Walker extended the definitions oft-norm (triangular norm) and t-conorm to L (subset of normal and convex functions of M), establishing the tr-norms and tr-conorms (according to the “restrictive axioms” given by them), and defined two families of binary operations on M and found that, under certain conditions, these operations are tr.-norms or tr.-conorms on L. In this paper, we introduce more general binary operations on M than those given by Walker and Walker and study which of the minimum conditions necessary for these operations satisfy each of the axioms of the tr-norm and tr-conorm. In particular, interesting results about the closure properties are obtained, and the main result of the paper provides sufficient conditions for the given operations to be tr.-norms or tr-conorms on L.

In a previous paper1, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open. The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

Trillas et al. (Soft Comput 1999;3(4):197–199 and In: Proc 18th Int Conf of the North American Fuzzy Information Processing Society (NAFIPS), New York;1999; pp 28–32) introduced the concepts of self-contradictory fuzzy set and contradictory fuzzy sets in an attempt to mark out when an inference process is not coherent. Later, contradiction was studied along the same lines in Cubillo and Castiñeira (In: Proc X Conf of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004), Perugia (Italy); (2004). 2180–2186) within the framework of Atanassov's intuitionistic fuzzy sets (AIFSs). The aim of this paper is to axiomatically model self-contradiction measures on AIFSs. After introducing some functions to measure negation-dependent or -independent degrees of self-contradiction of an AIFS in Castiñeira, Cubillo, and Torres (Mathware Soft-Comput 2006;13:139–156), a preliminary axiomatic model for measuring the self-contradiction of AIFSs was presented in Castiñeira et al. (In: Proc XI Conf of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006), Paris (France); 2006. pp 2391–2398). Being a very early model, it turned out to be incomplete. For this reason, this paper takes up the study started in Castiñeira et al. (2006) again with a view to filling up its gaps. Here, we present a more complete model that envisages the continuity of self-contradiction measures from a broader perspective. The concepts of semicontinuous and completely semicontinuous, from both below and above, are now introduced, and a classification result is shown. © 2009 Wiley Periodicals, Inc.

The paper focuses on the study of the contradiction between two Atanassov's intuitionistic fuzzy sets. First, taking into account some characterizations obtained in previous papers, some functions are defined in order to measure the degrees of contradiction. Besides the principal properties of these measures are pointed out. Finally, some results relating self-contradiction and contradiction between two Atanassov's intuitionistic fuzzy sets are achieved.

Some fuzzy reasoning systems base inference processes on fuzzy implication functions. Although there has been a great deal of work done on characterizing R- and S-implications, little is known about QM-implications in spite of their long history since they came to fuzzy logic by analogy with the quantum mechanic logic. This paper tackles the study of some characteristics of this type of operator. It focuses on the QM-implication operator both as an implication function and also as a T-conditional function, giving useful tools to characterize them. © 2000 John Wiley & Sons, Inc.

This paper investigates the use of functions other than t-norms to model the Modus Ponens rule in a fuzzy inference process. For that purpose, new definitions for fuzzy inference related concepts are suggested, that take into account the possibility of using a larger class of functions. In particular, the concept of "Modus Ponens generating function" is revisited, allowing to find out when and where (in which subset of the defined universe) an operator is able to generate the Modus Ponens scheme. In addition, given such an operator, the conditional relations that may be used along with it to model an inference process are found. These results are applied to some common operators, finding their Modus Ponens generation capacity as well as their corresponding residuated fuzzy conditionals. Finally, the relation between an operator's ability to describe the Modus Ponens rule and its conjunctive/disjunctive behaviour is also studied, by means of a series of sufficient and/or necessary conditions relating both concepts.

A new method is given for a finite reflexive relation with a fixed set of properties and a t-norm T, so that a different, but similar, T-transitive relation could be found. The transitive closure is greater than the given relation; however, the T-transitivized relation defined in this article is less than or equal to the original one, while reaching or maintaining, in this case, the T-conditionality. The algorithm also verifies if a relation is T-transitive. ©1999 John Wiley & Sons, Inc.

In his paper "Probabilistic Theories of Relations", Menger showed that, in some cases, Prod-Indistinguishabilities are the same as the antilogarithms of distances; later, Ovchinikov obtained a characterization for Prod-Indistinguishabilities. In this note, Menger's result is included in a larger frame to obtain more Prod-Indistinguishabilities from distances, and Ovchinnikov's characterization is translated to distances in order to reach a larger class of them.