Betweenness relations and Fuzzy betweenness relations in fuzzy metric spaces

TL;DR

This paper explores the construction and properties of betweenness and fuzzy betweenness relations in KM-fuzzy metric spaces, introducing two methods and proving their equivalence and transitivity properties.

Contribution

It introduces two novel construction methods for fuzzy betweenness relations in KM-fuzzy metric spaces and proves their equivalence and key properties.

Findings

Two construction methods for fuzzy betweenness relations are equivalent.

Fuzzy betweenness relations satisfy specific transitivity properties.

The family of betweenness relations forms a nest in KM-fuzzy metric spaces.

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.