Narrow Results

This paper is about the generalization of the concept of fuzzy metric. Here we introduce g-fuzzy metric space of order $n$. In particular case, it reduces to George- and Veeramani-type fuzzy metric and Q-fuzzy metric. Some non-trivial examples of g-fuzzy metric space are given and its topological structure, the notion of convergent sequence, Cauchy sequence, etc. are also studied.

To cover almost all known Lebesgue type integrals constructed by means of pseudo-operations, the (S, U)-integral based on a t-conorm S, a uninorm (or t-norm) U and an S-measure (decomposable measure) is introduced and its properties are studied. Also its relationship to aggregation operators is discussed.

This paper presents a characterization of a new family of operators. Namely, all non-decreasing, associative binary operators U: [0, 1]² → [0, 1] with a left (or right) neutral element e ∈ [0, 1], and such that they satisfy an additional hypothesis on continuity which is called here left (or right) pseudocontinuity.

A new approach to normalizing fuzzy sets is introduced where it is assumed that the normalization method is compatible with a given t-norm. In this context it is proved that the most usual ways to normalize fuzzy subsets correspond to the most common t-norms.

For a given fuzzy subset μ, the corresponding normalized fuzzy subset can be viewed as the distribution of μ conditioned on the (degree of) existence of its elements with maximal membership. From this view point we investigate the less specific normal fuzzy subset of X among the most similar fuzzy subsets to μ and the normal fuzzy subset generating the same fuzzy T-preorder as μ.

A characterization of all idempotent uninorms satisfying the distributive property is given. The special cases of left-continuous and right-continuous idempotent uninorms are presented separately and it is also proved that all idempotent uninorms are autodistributive. Moreover, all distributive pairs of idempotent uninorms (pairs U₁, U₂ such that U_l is distributive over U₂ and U₂ is distributive over U₁) are also characterized.

An axiomatic approach to scalar cardinalities of finite fuzzy sets involving t-norms and t-conorms is presented. A characterization theorem for these cardinalities is proved and it is also proved that some standard properties remain true. On the other hand, properties like finite additivity, valuation property or finite subadditivity depend on the t-norm and the t-conorm.

We consider the interval ]-1, 1[ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t. 0 are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

A new map (Λ_E) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of Λ_E is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φ_E and ψ_E) are also reviewed, in order to compare them with Λ_E. Interesting properties of the fixed points of φ_E and are studied. Among others, the fixed points of Λ_E (Fix(Λ_E)) are proved to be the maximal fuzzy points of (X, E) and the fixed points of coincide with the Image of Λ_E. An isometric embedding of X into Fix(Λ_E) is established and studied.

The α-migrative property for uninorms with different neutral elements is presented, and some general results are given. The case for idempotent uninorms is studied, characterizing those uninorms (from any of the main classes of uninorms), which are α-migrative over an idempotent uninorm. The solutions obtained generalize the results where both uninorms have the same neutral elements.

In this paper, we study t-norms and t-conorms on bounded lattices. We propose new methods for generating these operators, applicable on any bounded lattice M by use of the presence of a t-norm on [0_M, k] and a t-conorm on [k, 1_M] for an element k ∊ M\{0_M, 1_M}. In addition, some corresponding examples are provided for well understanding the structure of new t-norms and t-conorms.

Uninorms, as important generalizations of triangular norms and conorms, let the identity $e$ exist anywhere on a bounded lattice. In this paper, we focus on new characterizations of uninorms allowed to act on more general bounded lattices. In particular, we present several necessary and sufficient conditions to verify the construction approaches introduced by (Çaylı and Karaçal, Kybernetika 53 (2017) 394–417) and (Çaylı, Fuzzy Sets Syst. 395 (2020) 107–129), yielding a uninorm on bounded lattices.

Uninorms combining t-conorms and t-norms on bounded lattices have lately drawn extensive interest. In this article, we propose two ways for constructing uninorms on a bounded lattice with an identity element. They benefit from the appearance of the t-norm (resp. t-conorm) and the closure operator (resp. interior operator) on a bounded lattice. Additionally, we include some illustrative examples to highlight that our procedures differ from others in the literature.

We study some properties of De Morgan triplets. Firstly, we introduce submodular De Morgan triplets and we study its relationships with subdistributive ones. Moreover, we characterize them both in the strict and non strict archimedean cases. Secondly, we introduce the concepts of modularity, distributivity and (S, T)-distributivity degrees and we give some general results. Afterwards we apply these concepts to two particular cases, Lukasiewicz triplets and a kind of strict De Morgan triplets (Product triplets). For the Lukasiewicz ones we prove that all three degrees range over (0, 1/2]. For Product triplets, a recipe to calculate these degrees is given. In particular we present examples with distributivity degree r for all r ∈ (0, 1) and examples with (S, T)-distributivity and modularity degrees taking all values in (0, 3/4].

In this paper we deal with the idempotency equation H(x,x)=x for all x∈[0,1]. In particular we solve it for two special cases. First when H is a convex linear combination of a strict t-norm and its (1-j)-dual and second, when H is a convex linear combination of a special kind of aggregation functions F=<(f,N)> and its N-dual, being these aggregation functions, called L-representable aggregation functions, a kind of functions verifying a similar representation theorem to the classical representation theorem for non strict Archimedean t-norms.

This paper looks over a class of operators introduced in ([2]), called t–operators. Introduced in order to be applied to fuzzy preorders, their properties lead them to be also appropriate in some fields like aggregation problems and expert systems. We characterize these operators as a special combination of a t-norm and a t-conorm on [0, 1] in a similar way of uninorms in ([5]). We study duality and self duality on t–operators with respect to a strong negation N. We also give a classification of continuous t–operators through ordinal sums. Finally, we obtain from some t–operators (those idempotent) a special kind of E.A.F. by extending them to E=∪_n≥1[0,1]ⁿ.

The proofs of many factorization results for an intuitionistic fuzzy binary relation 〈ρ_μ,ρ_ν〉 involve dual proofs, one for ρ_μ with respect to a t-conorm ⊕ and one for ρ_ν with respect to a t-norm ⊗. In this paper, we show that one proof can be obtained from the other by considering ⊕ and ⊗ dual under an involutive fuzzy complement. We provide a series of singular proofs for commonly defined norms and conorms.

We generalize the definition of a fuzzy graph by replacing minimum in the basic definitions with an arbitrary $t$-norm. The reason for this is that some applications are better modeled with a $t$-norm other than minimum. We develop a measure on the susceptibility of trafficking in persons for networks by using a $t$-norm other than minimum. We also develop a connectivity index for a fuzzy network. In one application, a high connectivity index means a high susceptibility to trafficking. In the other application, we use a method called the eccentricity of an origin country to determine the susceptibility of a network to trafficking in persons. The models rest on the vulnerabilities and the government responses of countries to trafficking.

Fuzzy relations are fundamental in applications of fuzzy set theory and fuzzy logic. The entire literature on fuzzy relations as applied to fuzzy graph theory are based on Rosenfeld’s relations. Rosenfeld used minimum and maximum as the norm and conorm in his study of compositions of fuzzy relations. In this paper, we generalize fuzzy relations using arbitrary $t$-norms and $t$-conorms. Many of the results do not hold when minimum and maximum are replaced by an arbitrary norm and an arbitrary conorm. Reflexive, symmetric and transitive generalized fuzzy relations are also discussed and an application to human trafficking and illegal immigration is presented.

This study aims at investigating $L^B$-valued general fuzzy automata, for simplicity, $L^B$-valued GFA, with respect to their algebraic properties and based on t-norm/t-conorm and general implicators, where $L$ stands for residuated lattice and $B$ is a set of propositions about the GFA, in which its underlying structure is a complete infinitely distributive lattice. Specifically, we introduce the concepts of $L^B$-valued operators with t-norm and $L^B$-valued operators with t-conorm. We also examine the relationships between the $L^B$-valued operators with t-norm and $L^B$-valued operators with t-conorm. Finally, we introduce the concepts of $L^B$-valued operators with implicator and study the relationships between the $L^B$-valued operators with the implicator and the $L^B$-valued operators with t-norm/t-conorm. To clarify the notions and the results obtained in this study, some examples are submitted as well.

In our previous paper, it is shown that topology of G-fuzzy normed linear space is generated by two types of open balls: one is elliptic and the other is circular. In the theoretical aspect of functional analysis, will this type of exception happen or not? To address this problem in this paper, firstly, G-fuzzy bounded linear operators as well as G-fuzzy bounded linear functionals are defined which are the key elements of functional analysis. Then, operator G-fuzzy norms are introduced for both the cases using the idea of quasi-G-norm family. The definition of operator G-fuzzy norm is quite different from the existing operator fuzzy norm. Completeness of operator G-fuzzy norm is investigated. Lastly, Hahn-Banach theorem in G-fuzzy setting is studied using all the above concepts.