# Using Dual Double Fuzzy Semi-Metric to Study the Convergence
## 3 Double Fuzzy Semi-Metric
The mapping_ \(\zeta(x,y;u,v,\cdot)\) _is said to be symmetrically nondecreasing if and only if_ \(\zeta(x,y;u,v,t_{1})\geq\zeta(y,x;v,u,t_{2})\) _for_ \(t_{1}>t_{2}\)_._
* _The mapping_ \(\zeta(x,y;u,v,\cdot)\) _is said to be_ \(\triangleright\)_-ensim symmetrically nondecreasing if and only if_ \(\zeta(x,y;u,v,t_{1})\geq\zeta(y,x;u,v,t_{2})\) _for_ \(t_{1}>t_{2}\)_._
* _The mapping_ \(\zeta(x,y;u,v  
**Proposition 3**. : _Let \((X,M)\) be a fuzzy semi-metric space along with a t-norm \(*\). Given any fixed \(x,y,u,v\in X\), the double fuzzy semi-metric \(\zeta\) satisfies the following properties:_  
1. _Suppose that_ \(M\) _satisfies the_ \(\bowtie\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is nondecreasing._
2. _Suppose that_ \(M\) _satisfies the_ \(\triangleright\)_-triangle inequality or the_ \(\triangleleft\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is simultaneously nondecreasing, symmetrically nondecreasing,_ \(\triangleleft\)_-semisymmetrically nondecreasing, and_ \(\triangleright\)_-semisymmetrically nondecreasing._
3. _Suppose that_ \(M\) _satisfies the_ \(\triangleright\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is symmetrically nondecreasing._  
Proof. : Part (i) of Proposition 1 says that the mappings \(M(x,y,\cdot)\) and \(M(u,v,\cdot)\) from \([0,\infty)\) into \([0,1]\) are nondecreasing. According to the increasing property of t-norm, we conclude that the mapping \(\zeta(x,y;u,v,\cdot)\) from \([0,\infty)\) into \([0,1]\) is nondecreasing, which proves part (i). Part (ii) can be obtained from part (ii) of Proposition 1, and part (iii) can be obtained from part (iii) of Proposition 1. This completes the proof. By using the strictly increasing property of t-norm, the proof of Proposition 3 is still valid for obtaining the following results. **Proposition 4**. : _Let \((X,M)\) be a fuzzy semi-metric space along with a t-norm \(*\). Suppose that the t-norm satisfies the strictly increasing property. Given any fixed \(x,y,u,v\in X\), the double fuzzy semi-metric \(\zeta\) satisfies the following properties:_  
1. _Suppose that_ \(M\) _satisfies the strict_ \(\bowtie\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is increasing._
2. _Suppose that_ \(M\) _satisfies the strict_ \(\triangleright\)_-triangle inequality or the strict_ \(\triangleleft\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is simultaneously increasing, symmetrically increasing,_ \(\triangleleft\)_-semisymmetrically increasing, and_ \(\triangleright\)_-semisymmetrically increasing._
3. _Suppose that_ \(M\) _satisfies the strict_ \(\triangleright\)_-triangle inequality. Then the mapping_ \(\zeta(x,y;u,v,\cdot)\) _from_ \([0,\infty)\) _into_ \([0,1]\) _is symmetrically increasing._  
Let \((X,M)\) be a fuzzy semi-metric space. The motivation for considering the following two concepts can refer to Wu [16]. * \(M\) is said to satisfy the canonical condition if and only if: \[\zeta(x,z;u,w,t+s)\geq\zeta(x,y;u,v,t)*\zeta(z,y;w,v,s)\]3
* \(M\) is said to satisfy the rational condition if and only if: \[\zeta(x,z;u,w,t+s)\geq\zeta(x,y;u,v,t)*\zeta(z,y;w,v,s)\]4  
**Proposition 5**. : _Let \((X,M)\) be a fuzzy semi-metric space along with a t-norm \(*\)._  
1. _Suppose that_ \(M\) _satisfies the canonical condition. If the_ \(t\)_-norm_ \(*\) _is left-continuous at_ \(1\) _with respect to the first or second argument, then we have:_ \[\zeta(x,z;u,w,t+s)\geq\zeta(x,y;u,v,t)*\zeta(z,y;w,v,s)\]5 (8)
2. _Suppose that_ \(M\) _satisfies the rational condition. If the_ \(t\)_-norm_ \(*\) _is right-continuous at_ \(0\) _with respect to the first or second argument, then we have:_ \[\zeta(x,z;u,w,t+s)\geq\zeta(x,y;u,v,t)*\zeta(z,y;w,v,s)\]6 (9)Proof.