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Fuzzy betweenness relations in fuzzy metric spaces

Yu Zhong

TL;DR

Two different construction methods for fuzzy betweenness relations induced by a KM-fuzzy metric are introduced and one is directly obtained by using the implication operator and the other is through the corresponding nest of metrics of KM-fuzzy metrics.

Abstract

In this paper, we mainly discuss the constructions and the characteristics of betweenness relations and fuzzy betweenness relations in KM-fuzzy metric spaces. And the family of betweenness relations induced by a KM-fuzzy metric form a nest of betweenness relations.The main focus of this paper is to introduce two different construction methods for fuzzy betweenness relations induced by a KM-fuzzy metric.One of them is directly obtained by using the implication operator. The other is through the corresponding nest of metrics of KM-fuzzy metrics. Furthermore, we also show that the two types of fuzzy betweenness relations are the same, and they also satisfy the eight kinds of four-point transitivity properties and the six kinds of five-point transitivity properties.

Fuzzy betweenness relations in fuzzy metric spaces

TL;DR

Two different construction methods for fuzzy betweenness relations induced by a KM-fuzzy metric are introduced and one is directly obtained by using the implication operator and the other is through the corresponding nest of metrics of KM-fuzzy metrics.

Abstract

In this paper, we mainly discuss the constructions and the characteristics of betweenness relations and fuzzy betweenness relations in KM-fuzzy metric spaces. And the family of betweenness relations induced by a KM-fuzzy metric form a nest of betweenness relations.The main focus of this paper is to introduce two different construction methods for fuzzy betweenness relations induced by a KM-fuzzy metric.One of them is directly obtained by using the implication operator. The other is through the corresponding nest of metrics of KM-fuzzy metrics. Furthermore, we also show that the two types of fuzzy betweenness relations are the same, and they also satisfy the eight kinds of four-point transitivity properties and the six kinds of five-point transitivity properties.
Paper Structure (15 sections, 12 theorems, 30 equations, 5 figures)

This paper contains 15 sections, 12 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ternary relations
  4. Four-point transitivity properties of a ternary relation
  5. Five-point transitivity properties of a ternary relation
  6. Betweenness relations
  7. KM-fuzzy metric spaces
  8. Betweenness relations in KM-fuzzy metric spaces
  9. One-to-one correspondence between a KM-fuzzy metric and its nests of metrics
  10. Betweenness relations in KM-fuzzy metric spaces
  11. Fuzzy betweenness relations in KM-fuzzy metric spaces
  12. A fuzzy betweenness relation $\mathfrak{B}^{M}$ induced by a KM-fuzzy metric $M$
  13. A fuzzy betweenness relation $\mathfrak{B}^{\mathcal{D}^{M}}$ induced by its corresponding nest of metrics $\mathcal{D}^{M}$
  14. The relationship between fuzzy betweenness relations $\mathfrak{B}^{M}$ and $\mathfrak{B}^{\mathcal{D}^{M}}$
  15. Conclusions

Key Result

Theorem 3.2

Let $(X, M, \wedge)$ be a KM-fuzzy metric space. For any $a\in (0, 1)$, define a mapping $d_{a}^{M}: X\times X\rightarrow [0, \infty)$ by $\forall x, y\in X$, Then (1) $M(x, y, t)\leq a \Leftrightarrow d_{a}^{M}(x, y)\geq t$, i.e., $M (x, y, t)> a \Leftrightarrow d_{a}^{M}(x, y)<t$. (2) $d_{a}^{M}(x, y)=\bigwedge\{t\in [0, \infty) \mid M(x, y, t)> a\}$. (3) For any $a\in (0, 1)$, $d_{a}^{M}$ is

Figures (5)

  • Figure 1: A schematic diagram illustrating the four-point transitivity properties of ternary relations.
  • Figure 2: A schematic diagram illustrating the five-point transitivity properties of ternary relations.
  • Figure 3: One-to-one correspondence between a KM-fuzzy metric and its corresponding nests of metrics.
  • Figure 4: Constructions and Characterizations of betweenness relations in KM-fuzzy metric spaces.
  • Figure 5: Summary diagram of fuzzy betweenness relations induced by a KM-fuzzy metric.

Theorems & Definitions (45)

  • Definition 2.1: Bakri-2021Zedam-2018
  • Definition 2.2: Zedam-2020Zedam-2021-2
  • Definition 2.3: Zedam-2020Zedam-2021-2
  • Definition 2.4: Pérez-2019
  • Remark 1
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8: Frechet-1906
  • Definition 2.9: Klement-2000Schweizer-1983
  • ...and 35 more