Convergence in Fuzzy Semi-Metric Spaces

Abstract

The convergence using the fuzzy semi-metric and dual fuzzy semi-metric is studied in this paper. The infimum type of dual fuzzy semi-metric and the supremum type of dual fuzzy semi-metric are proposed in this paper. Based on these two types of dual fuzzy semi-metrics, the different types of triangle inequalities can be obtained. We also study the convergence of these two types of dual fuzzy semi-metrics.

Using Dual Double Fuzzy Semi-Metric to Study the Convergence

“…Given any fixed x, y, u, v ∈ X, the double fuzzy semi-metric ζ satisfies the following properties: Let (X, M) be a fuzzy semi-metric space. The motivation for considering the following two concepts can refer to Wu [16].…”

“…The convergence using fuzzy semi-metric has been studied in Wu [16], where the infimum type of dual fuzzy semi-metric is the function Γ ↓ (λ, •, •) : X × X → [0, +∞) defined by: Γ ↓ (λ, x, y) = inf {t > 0 : M(x, y, t) ≥ 1 − λ} , and the supremum type of dual fuzzy semi-metric is the function Γ ↑ (λ, •, •) : X × X → [0, +∞) defined by: Γ ↑ (λ, x, y) = sup {t > 0 : M(x, y, t) ≤ 1 − λ} .…”

“…Using the infimum and supremum types of dual fuzzy semi-metric Γ ↓ (λ, x, y) and Γ ↑ (λ, x, y), the convergence of sequences in (X, M) and the concept of Cauchy sequence in (X, M) have been studied in Wu [16]. In this paper, we study the extended convergence of sequences in (X, M) and the concept of joint Cauchy sequence in (X, M) using the infimum and supremum types of dual double fuzzy semi-metric Ψ ↓ (λ, x, y; u, v) and Ψ ↑ (λ, x, y; u, v).…”

“…16, we see that if ζ(x, y; u, v, t) ≤ 1 − λ, then t ≤ Ψ ↑ (λ, x, y; u, v).•Suppose that M satisfies the •-triangle inequality for • ∈ { , }. It is clear to see that the factΨ ↑ (λ, y, x; u, v) = +∞ implies t ≤ Ψ ↑ (λ, y, x; u, v).…”

“…In order to establish the triangle inequalities for the infimum type of dual double fuzzy semi-metric, we provide a useful lemma. [16]) Suppose that the t-norm * is left-continuous at 1 with respect to the first or second argument. For any a ∈ (0, 1) and any p ∈ N, there exists r ∈ (0, 1) such that:…”