# Using Dual Double Fuzzy Semi-Metric to Study the Convergence
## 1 Introduction  
The concept of fuzzy metric space proposed by [PERSON] and [PERSON] [1] was inspired by the Menger space that is a special kind of probabilistic metric space by referring to [PERSON] and [PERSON] [2; 3; 4], [PERSON] and [PERSON] [5], and [PERSON] et al. [6]. [PERSON] and [PERSON] [7] proposed another concept of fuzzy metric space by considering the membership degree of the distance between any two different points. [PERSON] and [PERSON] [8; 9] studied some properties of fuzzy metric spaces in the sense of [PERSON] and [PERSON] [1]. [PERSON] and [PERSON] [10; 11; 12] also extended the study of the properties of fuzzy metric spaces and fuzzy quasi-metric spaces in which the symmetric condition was not assumed. The Hausdorff topology induced by the fuzzy metric space was studied in [PERSON] [13]. In this paper, we shall propose the concept of double fuzzy-semi metric in fuzzy semi-metric space and study its convergent properties. The potential application for using the convergence of dual double fuzzy semi-metric is to study the new type of fixed point theorems in fuzzy semi-metric space by considering the Cauchy sequences, which will be the future research and may refer to the previous work of [PERSON] [14] for studying the common coincidence points and common fixed points in fuzzy semi-metric spaces. [PERSON] [15] studied the so-called fuzzy semi-metric space without assuming the symmetric condition. In the fuzzy semi-metric space \((X,M)\), the symmetric condition \(M(x,y,t)=M(y,x,t)\) for all \(x,y\in X\) and \(t>0\) is not assumed to be true. Therefore, four kinds of triangle inequalities should be considered. In order to obtain the new type of fixed point theorems in fuzzy semi-metric space, we need to study the convergence using dual double fuzzy semi-metric. Based on the concept of t-norm \(*\), we shall firstly define the double fuzzy semi-metric by considering the mapping \(\zeta:X^{4}\times[0,+\infty)\rightarrow[0,1]\) that is defined by:  
\[\zeta(x,y;u,v,t)=M(x,y,t)*M(u,v,t),\]  
where \(\zeta\) is called a double fuzzy semi-metric. The convergence using fuzzy semi-metric has been studied in [PERSON] [16], where the infimum type of dual fuzzy semi-metric is the function \(\Gamma^{\downarrow}(\lambda,\cdot,\cdot):X\times X\to[0,+\infty)\) defined by:  
\[\Gamma^{\downarrow}(\lambda,x,y)=\inf\left\{t>0:M(x,y,t)\geq 1-\lambda \right\},\]  
and the supremum type of dual fuzzy semi-metric is the function \(\Gamma^{\uparrow}(\lambda,\cdot,\cdot):X\times X\to[0,+\infty)\) defined by:  
\[\Gamma^{\uparrow}(\lambda,x,y)=\sup\left\{t>0:M(x,y,t)\leq 1-\lambda \right\}.\]  
In this paper, we shall consider the double fuzzy semi-metric \(\zeta\) to define the infimum and supremum types of dual double fuzzy semi-metric. The infimum type of dual double fuzzy semi-metric is the function \(\Psi^{\downarrow}(\lambda,\cdot,\cdot;\cdot,\cdot):X^{4}\to[0,+\infty)\) defined by:  
\[\Psi^{\downarrow}(\lambda,x,y;u,v)=\inf\left\{t>0:\zeta(x,y;u,v,t)\geq 1- \lambda\right\},\]  
and the supremum type of dual double fuzzy semi-metric is the function \(\Psi^{\uparrow}:X^{4}\to[0,+\infty)\) defined by:  
\[\Psi^{\uparrow}(\lambda,x,y;u,v)=\sup\left\{t>0:\zeta(x,y;u,v,t)\leq 1- \lambda\right\}.\]  
Using the infimum and supremum types of dual fuzzy semi-metric \(\Gamma^{\downarrow}(\lambda,x,y)\) and \(\Gamma^{\uparrow}(\lambda,x,y)\), 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 \(\Psi^{\downarrow}(\lambda,x,y;u,v)\) and \(\Psi^{\uparrow}(\lambda,x,y;u,v)\). As we mentioned above, these convergences will be used in the near future to establish the new types of fixed point theorems in fuzzy semi-metric space \((X,M)\). In Section 2, we review some basic properties of fuzzy semi-metric space that will be used for further discussion.