Using Dual Double Fuzzy Semi-Metric to Study the Convergence

Convergence using dual double fuzzy semi-metric is studied in this paper. Two types of dual double fuzzy semi-metric are proposed in this paper, which are called the infimum type of dual double fuzzy semi-metric and the supremum type of dual double fuzzy semi-metric. Under these settings, we also propose different types of triangle inequalities that are used to investigate the convergence using dual double fuzzy semi-metric.


Introduction
The concept of fuzzy metric space proposed by Kramosil and Michalek [1] was inspired by the Menger space that is a special kind of probabilistic metric space by referring to Schweizer and Sklar [2][3][4], Hadžić and Pap [5], and Chang et al. [6].Kaleva and Seikkala [7] proposed another concept of fuzzy metric space by considering the membership degree of the distance between any two different points.George and Veeramani [8,9] studied some properties of fuzzy metric spaces in the sense of Kramosil and Michalek [1].Gregori and Romaguera [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 Wu [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 Wu [14] for studying the common coincidence points and common fixed points in fuzzy semi-metric spaces.Wu [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 ∈ 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 ζ : X 4 × [0, +∞) → [0, 1] that is defined by: ζ(x, y; u, v, t) = M(x, y, t) * M(u, v, t), where ζ is called a double fuzzy semi-metric.
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).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.In Section 3, we introduce the concept of double fuzzy semi-metric and derive the related triangle inequalities.In Sections 4 and 5, the concepts of infimum and supremum types of dual double fuzzy semi-metric are proposed, and their convergent properties and triangle inequalities are studied.

Fuzzy Semi-Metric Space
Let X be a nonempty universal set, and let M be a mapping defined on X × X × [0, ∞) into [0, 1].Then (X, M) is called a fuzzy semi-metric space if and only if the following conditions are satisfied:

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For any x, y ∈ X, M(x, y, t) = 1 for all t > 0 if and only if x = y; • M(x, y, 0) = 0 for all x, y ∈ X with x = y.
We say that M satisfies the symmetric condition if and only if M(x, y, t) = M(y, x, t) for all x, y ∈ X and t > 0. We say that M satisfies the strongly symmetric condition if and only if M(x, y, t) = M(y, x, t) for all x, y ∈ X and t ≥ 0. Since the symmetric condition is not assumed to be true in fuzzy semi-metric space, four kinds of triangle inequalities called •-triangle inequality for • ∈ { , , , } were proposed by Wu [15].
Example 1.Let X be a universal set, and let d : X × X → R + satisfy the following conditions: • d(x, y) ≥ 0 for any x, y ∈ X; • d(x, y) = 0 if and only if x = y for any x, y ∈ X; • d(x, y) + d(y, z) ≥ d(x, z) for any x, y, z ∈ X.
Note that we do not assume d(x, y) = d(y, x).For example, let X = [0, 1].We define: Then d(x, y) = d(y, x) and the above three conditions are satisfied.Now we take t-norm * as a * b = ab and define: It is clear to see that M(x, y, t) = M(y, x, t) for t > 0, since d(x, y) = d(y, x).It is not hard to check that (X, M, * ) is a fuzzy semi-metric space satisfying the -triangle inequality.
The following interesting observations will be used in further study.Remark 1.Let (X, M) be a fuzzy semi-metric space.
In general, we have: • Suppose that M satisfies the -triangle inequality.Since: M(a, b, t 1 ) * M(c, b, t 2 ) ≤ min {M(a, c, t 1 + t 2 ), M(c, a, t 1 + t 2 )} , which implies: In general, we have: • Suppose that M satisfies the -triangle inequality.Since: In general, we have: • Suppose that M satisfies the -triangle inequality.Then: and: From Equation ( 4), we also have: which implies: by referring to Equation (5).In general, we have the following cases: (a) If p is even, then: If p is odd, then: Let (X, M) be a fuzzy semi-metric space.

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We say that M is nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(x, y, t 2 ) for t 1 > t 2 > 0.

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We say that M is increasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) > M(x, y, t 2 ) for t 1 > t 2 > 0.

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We say that M is symmetrically nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(y, x, t 2 ) for t 1 > t 2 > 0.

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We say that M is symmetrically increasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) > M(y, x, t 2 ) for t 1 > t 2 > 0.
The following interesting results were modified from Wu [15] using the similar argument, which will be used in further discussion.Proposition 1. (Wu [15]) Let (X, M) be a fuzzy semi-metric space.Then we have the following properties: (i) If M satisfies the -triangle inequality, then M is nondecreasing.If M satisfies the strict -triangle inequality, then M is increasing.(ii) If M satisfies the -triangle inequality or the -triangle inequality, then M is both nondecreasing and symmetrically nondecreasing.If M satisfies the strict -triangle inequality or the strict -triangle inequality, then M is both increasing and symmetrically increasing.(iii) If M satisfies the -triangle inequality, then M is symmetrically nondecreasing.If M satisfies the strict -triangle inequality, then M is symmetrically increasing.

Double Fuzzy Semi-Metric
Let (X, M) be a fuzzy semi-metric space along with a t-norm * .Given any four elements x, y, u, v ∈ X, recall that the value M(x, y, t) means the membership degree of the distance that is less than t between x and y, and the value M(u, v, t) means the membership degree of the distance that is less than t between u and v.In this case, we can define a value: which means the membership degree of the distance that is simultaneously less than t between x and y and between u and v.In general, instead of considering the min function, we shall use the t-norm.The formal definition is given below.Definition 1.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We define the mapping ζ : Then ζ is called a double fuzzy semi-metric.
Example 2. Continued from Example 1, we consider: If we take t-norm as a * b = a • b, then the double fuzzy semi-metric can be obtained as: The potential application for considering the double fuzzy semi-metric is to study the new type of fixed point theorems in fuzzy semi-metric space.Proposition 2. (Triangle Inequalities for Dual Fuzzy Semi-Metric) Let (X, M) be a fuzzy semi-metric space along with a t-norm * .Given any x, y, z, u, v, w ∈ X, we have the following properties: (i) Suppose that M satisfies the -triangle inequality.Then we have the inequality: for s, t > 0. (ii) Suppose that M satisfies the -triangle inequality.Then we have the inequality: for s, t > 0. (iii) Suppose that M satisfies the -triangle inequality.Then we have the inequality: for s, t > 0. (iv) Suppose that M satisfies the -triangle inequality.Then we have the inequality: for s, t > 0.
Definition 2. Let (X, M) be a fuzzy semi-metric space along with a t-norm * , and let ζ be a double fuzzy semi-metric given by: Given any fixed x, y, u, v ∈ X, we define the following concepts of monotonicity: Proposition 3. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .Given any fixed x, y, u, v ∈ X, the double fuzzy semi-metric ζ satisfies the following properties: (i) Suppose that M satisfies the -triangle inequality.Then the mapping ζ(x, y; u, v, •) from [0, ∞) into [0, 1] is nondecreasing.(ii) Suppose that M satisfies the -triangle inequality or the -triangle inequality.Then the mapping 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 ∈ X, the double fuzzy semi-metric ζ satisfies the following properties: (i) Suppose that M satisfies the strict -triangle inequality.Then the mapping ζ(x, y; u, v, •) from [0, ∞) into [0, 1] is increasing.(ii) Suppose that M satisfies the strict -triangle inequality or the strict -triangle inequality.Then the mapping ζ(x, y; u, v, •) from [0, ∞) into [0, 1] is simultaneously increasing, symmetrically increasing, -semisymmetrically increasing, and -semisymmetrically increasing.(iii) Suppose that M satisfies the strict -triangle inequality.Then the mapping ζ(x, y; u, v, •) from [0, ∞) 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].

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M is said to satisfy the canonical condition if and only if: lim t→+∞ M(x, y, t) = 1 for any fixed x, y ∈ X.
• M is said to satisfy the rational condition if and only if: lim t→0+ M(x, y, t) = 0 for any fixed x, y ∈ X.
Proposition 5. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .
(i) 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: (ii) 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: Proof.To prove part (i), the canonical condition says that: The left-continuity of t-norm * at 1 also says that: To prove part (ii), the rational condition says that: The right-continuity of t-norm * at 0 also says that: This completes the proof.

Convergence Based on the Infimum
From Definition 1, we see that the double fuzzy semi-metric ζ is a mapping from X 4 × [0, ∞) into [0, 1].Here, we shall consider its dual sense by considering the mapping from (0 The formal definition is given below.Definition 3. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Given any fixed x, y, u, v ∈ X and any fixed λ ∈ (0, 1], we consider the following set: In this case, the mapping where: We also have: The potential application of dual double fuzzy semi-metric will be used to study the fixed point theorems in fuzzy semi-metric space.However, we first need to claim that the set which contradicts Equation ( 8).Therefore, Definition 3 is well-defined and Π ↓ (λ, x, y; u, v) = ∅.

Remark 2.
The following observations will be useful for further discussion.
• For any λ ∈ (0, 1], we have: and: • Given any fixed x, y, u, v ∈ X, if λ 1 > λ 2 , then: Proposition 6.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Given any fixed x, y, u, v ∈ X, suppose that the following conditions are satisfied: and: Then we have: Moreover, the following limit exists: Proof.
Proposition 10.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Given any fixed x, y, u, v ∈ X and λ ∈ (0, 1), the following statements hold true: Since can be any positive real number, we must have t ≤ Ψ ↓ (λ, y, x; u, v).We can similarly obtain another inequality using the fact of the mapping ζ(x, y; u, v, •) being -semisymmetrically nondecreasing.

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Suppose that M satisfies the Since can be any positive real number, we must have t ≤ Ψ ↓ (λ, y, x; v, u).
In order to establish the triangle inequalities for the infimum type of dual double fuzzy semi-metric, we provide a useful lemma.Lemma 1. (Wu [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: Theorem 1. (Triangle Inequalities for Dual Double Fuzzy Semi-Metric) Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Given any fixed µ ∈ (0, 1] and any fixed and distinct x 1 , x 2 , • • • , x p , y 1 , y 2 , • • • , y p ∈ X, we have the following inequalities: (i) Suppose that M satisfies the -triangle inequality.Then, there exists λ ∈ (0, 1), satisfying: Ψ ↓ (µ, x p , x 1 ; y 1 , y p ) ≤ Ψ ↓ (λ, x p , x p−1 ; y p−1 , y p ) + Ψ ↓ (λ, x p−1 , x p−2 ; y p−2 , y p−1 ) x 1 ; y 2 , y 1 ).
By taking → 0+, we obtain the desired inequality (Equation ( 25)).Since the other inequalities can be similarly obtained, we omit the details.
The above argument is still valid to obtain part (ii) by referring the second observation of Remark 1. Further, we can use the third observation of Remark 1 to obtain part (iii).Finally, part (iv) can be obtained by referring to the fourth observation of Remark 1.This completes the proof.
Let (X, M) be a fuzzy semi-metric space, and let {x n } ∞ n=1 be a sequence in X.We write x n M −→ x as n → ∞ if and only if: lim n→∞ M(x n , x, t) = 1 for all t > 0.
We also write x n M −→ x as n → ∞ if and only if: The main convergence theorem is presented below.We first provide a useful lemma.
A contradiction occurs.Therefore, we must have a > k.We can similarly show that b > k.This completes the proof.Theorem 2. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Let {x n } ∞ n=1 and {y n } ∞ n=1 be two sequences in X.Then we have the following properties: −→ y as n → ∞ if and only if Ψ ↓ (λ, x n , x; y n , y) → 0 as n → ∞ for all λ ∈ (0, 1).
Proof.For any fixed λ ∈ (0, 1), using Lemma 1, it follows that there exists λ 0 ∈ (0, 1), satisfying: We just prove the first case, since the other cases can be similarly obtained.Suppose that M(x n , x, t) → 1 and M(y n , y, t) → 1 as n → ∞ for all t > 0.Then, given any t > 0 and δ > 0, there exists n t,δ .Therefore, given any ∈ (0, 1), there exists n ∈ N, satisfying: for n ≥ n .We also have: for n ≥ n .The increasing property of t-norm says that: The definition of Ψ ↓ says that: for n ≥ n .This shows that Ψ ↓ (λ, x n , x; y n , y) → 0 as n → ∞.

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The sequence {x n } ∞ n=1 is said to be a >-Cauchy sequence in a metric sense if and only if, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N, satisfying M(x m , x n , t) > 1 − r for all pairs (m, n) of integers m and n with m > n ≥ n r,t .

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The sequence {x n } ∞ n=1 is said to be a <-Cauchy sequence in a metric sense if and only if, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N, satisfying M(x n , x m , t) > 1 − r for all pairs (m, n) of integers m and n with m > n ≥ n r,t .

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The sequence {x n } ∞ n=1 is said to be a Cauchy sequence in a metric sense if and only if, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N satisfying M(x m , x n , t) > 1 − r and M(x n , x m , t) > 1 − r for all pairs (m, n) of integers m and n with m, n ≥ n r,t and m = n.Definition 5. Let (X, M) be a fuzzy semi-metric space such that M satisfies the canonical condition, and let {x n } ∞ n=1 and {y n } ∞ n=1 be two sequences in X.

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Given any fixed λ ∈ (0, 1), the sequences {x n } ∞ n=1 and {y n } ∞ n=1 are said to be the joint (λ, >, <)-Cauchy sequences with respect to Ψ ↓ if and only if, given any > 0, there exists Given any fixed λ ∈ (0, 1), the sequences {x n } ∞ n=1 and {y n } ∞ n=1 are said to be the joint (λ, <, >)-Cauchy sequences with respect to Ψ ↓ if and only if, given any > 0, there exists Given any fixed λ ∈ (0, 1), the sequences {x n } ∞ n=1 and {y n } ∞ n=1 are said to be the joint (λ, <, <)-Cauchy sequences with respect to Ψ ↓ if and only if, given any > 0, there exists n ,λ ∈ N such that m > n ≥ n ,λ implies Ψ ↓ (λ, x n , x m ; y n , y m ) < .Theorem 3. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the canonical condition, and that the t-norm * is left-continuous at 1 with respect to the first or second argument.Let {x n } ∞ n=1 and {y n } ∞ n=1 be two sequences in X.Then, we have the following properties: (i) {x n } ∞ n=1 and {y n } ∞ n=1 are two >-Cauchy sequences in a metric sense if and only if {x n } ∞ n=1 and {y n } ∞ n=1 are the joint (λ, >, >)-Cauchy sequences with respect to Ψ ↓ for any λ ∈ (0, 1).
Proof.It suffices to just prove part (i), since the other cases can be similarly obtained.Suppose that {x n } ∞ n=1 and {y n } ∞ n=1 are >-Cauchy sequences.Then, given any t > 0 and δ > 0, there exists n t,δ ∈ N such that m > n ≥ n t,δ implies M(x m , x n , t) > 1 − δ and M(y m , y n , t) > 1 − δ.Now, given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies: The increasing property of t-norm says that: Further, by referring to the definition of Ψ ↓ , we obtain: Conversely, using the assumption, for any fixed t > 0 and given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies Ψ( /2, x m , x n ; y m , y n ) < t.Using Proposition 9, we obtain: and {y n } ∞ n=1 are >-Cauchy sequences.This completes the proof.

Convergence Based on the Supremum
Using the infimum and assuming the canonical condition, the infimum type of dual double fuzzy semi-metric was proposed in the previous section.In this section, we shall consider the supremum to propose the so-called supremum type of dual double fuzzy semi-metric.
Recall that the purpose for considering the canonical condition is to guarantee the infimum type of dual fuzzy semi-metric space to be well-defined.Now, we shall consider the rational condition to guarantee the supremum type of dual fuzzy semi-metric space to be well-defined.The formal definition is given below.Definition 6.Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the rational condition, and that the t-norm * is right-continuous at 0 with respect to the first or second argument.Given any fixed x, y, u, v ∈ X with x = y or u = v and any fixed λ ∈ [0, 1), we consider the following set: which will be used to define a function Ψ ↑ : X 4 → [0, +∞) by: The mapping Π ↑ from (0, 1] × X 4 into [0, ∞) is called the supremum type of dual double fuzzy semi-metric.Example 6. Continued from Example 1, we have: where: We also have: For any x = y or u = v, we need to claim that the set Π ↑ (λ, x, y; u, v) is nonempty.Suppose that Π ↑ (λ, x, y; u, v) = ∅.The definition says that ζ(x, y; u, v, t) > 1 − λ for all t > 0. Therefore, we obtain: which contradicts Equation ( 9).This says that Definition 6 is well-defined, which also says that Ψ ↑ (λ, x, y; u, v) > 0. We also have: Proposition 12. Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the rational condition, and that the t-norm * is right-continuous at 0 with respect to the first or second argument.Given any fixed x, y, u, v ∈ X with x = y or u = v, suppose that Ψ ↑ (λ, x, y; u, v) = +∞.Then, the following statements hold true: (i) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.Then we have ζ(x, y; u, v, t) ≤ 1 − λ for all t > 0. (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , }.Then we have ζ(y, x; u, v, t) ≤ 1 − λ and ζ(x, y; v, u, t) ≤ 1 − λ for all t > 0. (iii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.Then we have ζ(y, x; v, u, t) ≤ 1 − λ for all t > 0.
Proposition 13.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the rational and canonical conditions, and that the t-norm * is right-continuous at 0 and left-continuous at 1 with respect to the first or second argument.Then, given any fixed x, y, u, v ∈ X with x = y or u = v, we have Ψ ↑ (λ, x, y; u, v) < +∞ for λ ∈ (0, 1).
Proposition 14.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .Assume that M satisfies the canonical and rational conditions.We also assume that the t-norm * is left-continuous at 1 and right-continuous at 0 with respect to the first or second argument, and that the t-norm * also satisfies the strictly increasing property.For any fixed x, y, u, v ∈ X with x = y or u = v, the following statements hold true: (i) Suppose that M satisfies the strict •-triangle inequality for • ∈ { , , }.Then we have: for each λ ∈ (0, 1).(ii) Suppose that M satisfies the strict •-triangle inequality for • ∈ { , }.Then we have: for each λ ∈ (0, 1).(iii) Suppose that M satisfies the strict •-triangle inequality for • ∈ { , , }.Then we have: for each λ ∈ (0, 1).
To prove part (iii), parts (ii) and (iii) of Proposition 10 say that t ≤ Ψ ↓ (λ, y, x; v, u), which implies Ψ ↑ (λ, x, y; u, v) − < Ψ ↓ (λ, y, x; v, u).Since can be any positive real number, we obtain the desired inequality.This completes the proof.Proposition 15.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the rational condition, and that the t-norm * is right-continuous at 0 with respect to the first or second argument.For any fixed x, y, u, v ∈ X with x = y or u = v, and any fixed λ ∈ (0, 1), we assume Ψ ↑ (λ, x, y; u, v) < +∞.
According to Definition 5, we can similarly define the concepts of four types of the joint Cauchy sequences with respect to Ψ ↑ .We omit the details.Theorem 6.Let (X, M) be a fuzzy semi-metric space along with a t-norm * .We also assume that M satisfies the rational condition, and that the t-norm * is right-continuous at 0 with respect to the first or second argument.Let {x n } ∞ n=1 and {y n } ∞ n=1 be two sequences in X.Then we have the following properties: (i) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.Then the following statements hold true: • {x n } ∞ n=1 and {y n } ∞ n=1 are two >-Cauchy sequences in a metric sense if and only if {x n } ∞ n=1 and {y n } ∞ n=1 are the joint (λ, >, >)-Cauchy sequences with respect to Ψ ↑ for any λ ∈ (0, 1).

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{x n } ∞ n=1 and {y n } ∞ n=1 are two <-Cauchy sequences in a metric sense if and only if {x n } ∞ n=1 and {y n } ∞ n=1 are the joint (λ, <, <)-Cauchy sequences with respect to Ψ ↑ for any λ ∈ (0, 1).(ii) Suppose that M satisfies the -triangle inequality.Then the following statements hold true:
Proof.For any fixed λ ∈ (0, 1), using Lemma 1, it follows that there exists λ 0 ∈ (0, 1), satisfying: (1 To prove part (i), we just prove the first case, since the other cases can be similarly obtained.Suppose that {x n } ∞ n=1 and {y n } ∞ n=1 are >-Cauchy sequences in a metric sense.Therefore, given any t > 0 and δ > 0, there exists n t,δ ∈ N such that m > n ≥ n t,δ implies M(x m , x n , t) > 1 − δ and M(y m , y n , t) > 1 − δ.Now, given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies: The increasing property of t-norm says that: Further, the first result of part (ii) of Proposition 17 says that: To prove the converse, from the assumption, we see that for any fixed t > 0 and given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies Ψ ↑ ( , x m , x n ; y m , y n ) < t.Therefore, using the first result of part (i) of Proposition 16, we obtain ζ(x m , x n ; y m , y n , t) > 1 − for m > n ≥ n , i.e.,: M (x m , x n , t) * M (y m , y n , t) > 1 − , for m > n ≥ n .Lemma 2 says that: M (x m , x n , t) > 1 − and M (y m , y n , t) > 1 − , for m > n ≥ n .This shows that {x n } ∞ n=1 and {y n } ∞ n=1 are >-Cauchy sequences in a metric sense.To prove part (ii), the first result to the fourth result can be similarly obtained by the third result of part (ii) of Proposition 17.For proving the fifth result, using the assumption, we see that for any fixed t > 0 and given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies Ψ ↑ ( , x m , x n ; y m , y n ) < t.The third result of part (i) of Proposition 16 says that ζ(x n , x m ; y n , y m , t) > 1 − for m > n ≥ n .Therefore, we obtain: M (x n , x m , t) * M (y n , y m , t) > 1 − , for m > n ≥ n .This shows that {x n } ∞ n=1 and {y n } ∞ n=1 are <-Cauchy sequences in a metric sense.The remaining three results can be similarly obtained.This completes the proof.

Example 3 .
Continued from Example 1, it is not hard to check that M satisfies both the canonical and rational conditions.Suppose that we take t-norm as a * b = a • b.Then Proposition 5 says that: lim t→+∞ ζ(x, y; u, v, t) = 1 and lim t→0+ ζ(x, y; u, v, t) = 0.

Example 4 .
called the infimum type of dual double fuzzy semi-metric.Continued from Example 2, we have:

Lemma 2 .
Let * be a t-norm.If a * b > k then a > k and b > k.Proof.Since b ≤ 1, the increasing property and boundary condition show that b * k ≤ 1 * k = k.Suppose that a ≤ k.Then we have a * b ≤ k * b and: ) ≥ ζ(x, y; v, u, t 2 ) for t 1 > t 2 .The mapping ζ(x, y; u, v, •) is said to be -semisymmetrically increasing if and only if ζ(x, y; u, v, t 1 ) > ζ(x, y; v, u, t 2 ) for t 1 < t 2 .