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.

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.

Following the concept of l-fuzzy convergent sequence, l-fuzzy closed, l-fuzzy complete, l-fuzzy compact sets in fuzzy normed linear spaces, in this paper, we study some results in finite-dimensional fuzzy strong $\varphi$-b-normed linear spaces and extend the most momentous Riesz Lemma.

In this paper, concept of fuzzy continuous operator, fuzzy bounded linear operator are introduced in fuzzy strong $\varphi$-b-normed linear spaces and their relations are studied. Idea of operator fuzzy norm is developed and completeness of BF(X,Y) is established.

This paper consists of the proof of the equivalence between two fuzzy strong $\varphi$-b-norms on finite-dimensional linear space and establish a Banach-type contraction theorem in this new setting.