Convergence in Fuzzy Semi-Metric Spaces

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.


Introduction
Let * be a t-norm, and let M be a mapping defined on X × X × [0, ∞) into [0, 1].The three-tuple (X, M, * ) is called a fuzzy metric space if and only if some required conditions are satisfied.For different researchers, the required conditions are slightly different.However, the following symmetric condition and triangle inequality will be included
George and Veeramani [1,2] studied some properties of fuzzy metric spaces.Gregori and Romaguera [3][4][5] also extended this to study the properties of fuzzy metric spaces and fuzzy quasi-metric spaces.The Hausdorff topology induced by the fuzzy metric space was studied in Wu [6].Gregori and Romaguera [4] proposed the fuzzy quasi-metric spaces in which the symmetric condition was not assumed.Wu [7] studied the so-called fuzzy semi-metric space without assuming the symmetric condition in which four forms of triangle inequalities are considered.The common coincidence points and common fixed points in fuzzy semi-metric spaces were also studied in Wu [8].
Since the symmetric condition is not satisfied in the fuzzy semi-metric space, three kinds of limit concepts will be considered in this paper.Based on these limit concepts, we shall study the metric convergence in this paper.On the other hand, we also propose the concepts of the dual fuzzy semi-metric.We shall separately study the infimum type of dual fuzzy semi-metric and the supremum type of dual fuzzy semi-metric.Under these settings, we shall also investigate the convergence of the dual fuzzy semi-metric.The potential application for using the convergence of dual fuzzy semi-metric is to study the fixed point theorems in fuzzy semi-metric space by considering the Cauchy sequences, which will be the future research.
This paper is organized as follows.In Section 2, the basic properties of fuzzy semi-metric space are presented that will be used for the further discussion.In Section 3, since the symmetric condition is not necessarily satisfied, we study the metric convergence based on the different concepts of limits.In Section 4, we propose the concept of the infimum type of dual fuzzy semi-metric and study its convergent properties.Four different types of triangle inequalities can be obtained for this kind of dual fuzzy semi-metric.In Section 5, we also propose the concept of the supremum type of dual fuzzy semi-metric and study its convergent properties and triangle inequalities.

Fuzzy Semi-Metric Space
In the sequel, we shall define the concept of fuzzy semi-metric space without considering the symmetric condition.Due to lacking symmetry, the concept of the triangle inequality should be carefully interpreted.First of all, the so-called fuzzy semi-metric space is defined below.Definition 1.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 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.
From the first condition, we see that the value of M(x, x, 0) is free and M(x, x, t) = 1 for all t > 0. Since the value M(x, y, t) is interpreted as the membership degree of the distance that is less than t between x and y, the value M(x, x, t) = 1 for all t > 0 means that the distance that is less than t > 0 between x and x is always true.Regarding the second condition, the value M(x, y, 0) = 0 for x = y can be similarly realized such that the distance that is less than zero between two distinct elements x and y is impossible.
In order to consider the triangle inequalities, we need to introduce the concept of the t-norm.We say that the function * : [0, 1] × [0, 1] → [0, 1] is a t-norm if and only if the following conditions are satisfied Since the symmetric condition is not assumed to be true in fuzzy semi-metric space, four kinds of triangle inequalities proposed by Wu [7] are shown below.Definition 2. Let X be a nonempty universal set; let * be a t-norm; and let M be a mapping defined on X × X × [0, ∞) into [0, 1].

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
We say that M satisfies the strict -triangle inequality if and only if the inequality "≤" is replaced by the strict inequality "<".

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied M(x, y, t) * M(z, y, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
We say that M satisfies the strict -triangle inequality if and only if the inequality "≤" is replaced by the strict inequality "<".

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied M(y, x, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
We say that M satisfies the strict -triangle inequality if and only if the inequality "≤" is replaced by the strict inequality "<".

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied M(y, x, t) * M(z, y, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
We say that M satisfies the strict -triangle inequality if and only if the inequality "≤" is replaced by the strict inequality "<".
We say that M satisfies the strong •-triangle inequality for • ∈ { , , , } when s, t > 0 is replaced by s, t ≥ 0. The concept of the strong strict •-triangle inequality for • ∈ { , , , } can be similarly defined.
The following observations will be used in the further study.
Remark 1.Let (X, M) be a fuzzy semi-metric space.

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Suppose that M satisfies the -triangle inequality.Then In general, we have • Suppose that M satisfies the -triangle inequality.Since In general, we have • Suppose that M satisfies the -triangle inequality.Since In general, we have • Suppose that M satisfies the -triangle inequality.Then From ( 4), we also have by referring to (5).In general, we have the following cases.
(a) If p is even, then Definition 3. 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 results are modified from Wu [7] by using a similar argument, which will be used in the further discussion.Proposition 1.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.
Definition 4. Let (X, M) be a fuzzy semi-metric space.

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We say that M is left-continuous with respect to the distance at t 0 > 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 We say that M is left-continuous with respect to the distance on (0, ∞) if and only if the mapping M(x, y, •) is left-continuous on (0, ∞) for any fixed x, y ∈ X.

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We say that M is right-continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 We say that M is right-continuous with respect to the distance on [0, ∞) if and only if the mapping M(x, y, •) is left-continuous on [0, ∞) for any fixed x, y ∈ X.

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We say that M is continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists We say that M is continuous with respect to the distance on [0, ∞) if and only if the mapping M(x, y, •) is continuous on [0, ∞) for any fixed x, y ∈ X.

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We say that M is symmetrically left-continuous with respect to the distance at t 0 > 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t 0 − t < δ implies |M(x, y, t) − M(y, x, t 0 )| < .We say that M is symmetrically left-continuous with respect to the distance on (0, ∞) if and only if it is symmetrically left-continuous with respect to the distance at each t > 0.

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We say that M is symmetrically right-continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t − t 0 < δ implies |M(x, y, t) − M(y, x, t 0 )| < .We say that M is symmetrically right-continuous with respect to the distance on [0, ∞) if and only if it is symmetrically right-continuous with respect to the distance at each t ≥ 0.

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We say that M is symmetrically continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that |t − t 0 | < δ implies |M(x, y, t) − M(y, x, t 0 )| < .We say that M is symmetrically continuous with respect to the distance on [0, ∞) if and only if it is symmetrically continuous with respect to the distance at each t ≥ 0.
Example 1.Let X be a universal set.We consider a mapping d : X × X → R + satisfying 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.
We do not assume d(x, y) = d(y, x).For example, we can take X = [0, 1] and define Then, d(x, y) = d(y, x) and the above three conditions are satisfied.Now, we take t-norm * as the product a * b = ab and define Then, (X, M, * ) is a fuzzy semi-metric space satisfying the -triangle inequality.For any fixed x, y ∈ X, it is obvious that M(x, y, t) is continuous on (0, ∞).Therefore, M is continuous with respect to the distance on (0, ∞).
The following results are from Wu [7].Especially, Part (ii) is modified from Wu [7] by using the similar argument.Proposition 2. Let (X, M) be a fuzzy semi-metric space such that the •-triangle inequality is satisfied for • ∈ { , , }.Then, we have the following properties.
(i) Suppose that M is left-continuous or symmetrically left-continuous with respect to the distance at t > 0.
Then, M(x, y, t) = M(y, x, t).In other words, if M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞), then M satisfies the symmetric condition.(ii) Suppose that M is right-continuous or symmetrically right-continuous with respect to the distance at t > 0.Then, M(x, y, t) = M(y, x, t).In other words, if M is right-continuous or symmetrically right-continuous with respect to the distance on (0, ∞), then M satisfies the symmetric condition.
From Proposition 2, if M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞), or right-continuous and or symmetrically right-continuous with respect to the distance on (0, ∞], then we can just consider the -triangle inequality.The following results are modified from Wu [7] by using a similar argument.Proposition 3. Let (X, M) be a fuzzy semi-metric space.
(i) Suppose that M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞).
If M(x, x, 0) = 1 or M(x, x, 0) = 0 for any x ∈ X, then M satisfies the •-triangle inequality if and only if M satisfies the strong •-triangle inequality for • ∈ { , , }. (ii) Suppose that M is right-continuous or symmetrically right-continuous with respect to the distance on [0, ∞).Then, M satisfies the •-triangle inequality if and only if M satisfies the strong •-triangle inequality for • ∈ { , , }.

Metric Convergence
Since the symmetric condition is not satisfied in the fuzzy semi-metric space, three kinds of limit concepts will also be considered in this paper by referring to Wu [7].
Let (X, d) be a metric space.If the sequence {x n } ∞ n=1 in (X, d) converges to x, i.e., d(x n , x) → 0 as n → ∞, then it is denoted by x n d −→ x as n → ∞.In this case, we also say that x is a d-limit of the sequence {x n } ∞ n=1 .From Wu [7], the limits based on the fuzzy semi-metric M are given below.
Definition 5. Let (X, M) be a fuzzy semi-metric space, and let {x n } ∞ n=1 be a sequence in X.

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We write x n M −→ x as n → ∞ if and only if lim n→∞ M(x n , x, t) = 1 for all t > 0.
In this case, we call x a M -limit of the sequence {x n } ∞ n=1 .

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We write x n M −→ x as n → ∞ if and only if lim n→∞ M(x, x n , t) = 1 for all t > 0.
In this case, we call x a M -limit of the sequence {x n } ∞ n=1 .

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We write x n M −→ x as n → ∞ if and only if In this case, we call x an M-limit of the sequence {x n } ∞ n=1 .
The uniqueness of limits obtained from Wu [7] are shown below.
Proposition 4. Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous at one with respect to the first or second component, and let {x n } ∞ n=1 be a sequence in X.
(i) Suppose that M satisfies the -triangle inequality or the -triangle inequality.Then, we have the following properties.
Therefore, we obtain which implies x = y.This verifies Part (i) of Proposition 4.
In the sequel, we are going to present the different kinds of convergence of real-valued sequence We first provide some useful lemmas.Lemma 1.Let * be a t-norm.We have the following properties.Proof.To prove Part (i), we note that there exist two increasing sequences {p n } ∞ n=1 and {r n } ∞ n=1 such that p n ↑ a and r n ↑ b satisfying p n ≤ a n and r n ≤ b n .According to the concept of commutativity of the t-norm, we see that the t-norm is left-continuous with respect to each component.Given any > 0, using the left-continuity of the t-norm at b with respect to the second component, there exists Furthermore, using the left-continuity of the t-norm at a with respect to the first component, there exists Using the increasing property of the t-norm, for m ≥ n 1 and n ≥ n 0 , we have This shows the desired convergence.Part (ii) can be similarly proven, and the proof is complete.
(ii) Suppose that the t-norm * is left-continuous with respect to the first or second component.

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We have (iii) Suppose that the t-norm * is right-continuous with respect to the first or second component.
(vi) Suppose that the t-norm * is left-continuous with respect to the first or second component.Let {c n } n=1 be another increasing sequence in [0 Proof.To prove Part (i), by the increasing property of the t-norm, we immediately have for all k, which proves the desired inequalities.
To prove Part (ii), given any > 0, there exists m, n ∈ N such that The increasing property of the t-norm says that Since the t-norm is left-continuous with respect to each component, by taking → 0, we obtain the desired inequality.Now, according to the inequalities in Part (i), we have To prove Part (iii), given any > 0, there exists m, n ∈ N such that The increasing property of the t-norm says that Since the t-norm is right-continuous with respect to each component, by taking → 0, we obtain the desired inequality.
To prove Part (iv), according to the inequalities in Part (i), we have To prove Part (v), we also have To prove Part (vi), since 0 ≤ a n ≤ 1 and 0 ≤ b n ≤ 1 for all n and a n → 1 and b n → 1 as n → ∞, we see that sup Given any > 0, there exists m, n, r ∈ N such that Since a n → 1, b n → 1 and {c n } n=1 is an increasing sequence, there exists n 0 ∈ N such that a m ≤ a n 0 , b n ≤ b n 0 and c r ≤ c n 0 .The increasing property of the t-norm says that Since the t-norm is left-continuous with respect to each component, by taking → 0, we obtain Then, we obtain the desired equality.On the other hand, we have This completes the proof.
Proposition 5. Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞).Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality, and that t n → t

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If M is left-continuous with respect to the distance at t • , then • If M is symmetrically left-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the -triangle inequality and that t n → t (iii) Suppose that M satisfies the -triangle inequality and that t n → t If M is left-continuous or symmetrically left-continuous with respect to the distance at t • , then (iv) Suppose that M satisfies the -triangle inequality, and that t n → t Proof.To prove Part (i), given any sufficiently small > 0 with < t • /2, there exists n 0 ∈ N such that t • − t n < for all n ≥ n 0 .Using Part (i) of Proposition 1 and (1) in the first observation of Remark 1, we have which also say that the sequences {M(x n , x • , 2 )} ∞ n=1 and {M(y • , y n , 2 )} ∞ n=1 converge to one from the left, since each element of the sequences is less than one.The existence of the limit in the right-hand side of ( 8) is guaranteed by applying Part (i) of Lemma 1.Then, we obtain To prove Part (ii), using Part (ii) of Proposition 1 and (2) in the second observation of Remark 1, we have and From Part (i) of Proposition 2, we also have To prove Part (iii), using Part (ii) of Proposition 1 and (3) in the third observation of Remark 1, we have and From Part (i) of Proposition 2, we also have ).Therefore, we can similarly obtain the desired results.
To prove Part (iv), using Part (iii) of Proposition 1 and ( 4) and ( 8) in the fourth observation of Remark 1, we have From Part (i) of Proposition 2, we also have ).Therefore, we can similarly obtain the desired results, and the proof is complete.
In Proposition 5, we remark that M(x n , y n , t n ) = M(y n , x n , t n ) in general for n ∈ N. Proposition 6.Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞).Assume that the following inequality is satisfied for any sequences {a n } ∞ n=1 and {b n } ∞ n=1 in [0, 1].Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality and that t n → t

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If M is right-continuous with respect to the distance at t • , then • If M is symmetrically right-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the -triangle inequality and that t n → t If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then (iii) Suppose that M satisfies the -triangle inequality and that t n → t If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then Proof.To prove Part (i), given any fixed > 0, there exists n 0 ∈ N such that t n − t • < for all n ≥ n 0 .Using Part (i) of Proposition 1 and the first observation of Remark 1, we have for all n ≥ n 0 .To prove the first case of Part (i), we have (by Part (iv) of Lemma 2) If M is right-continuous with respect to the distance at t • , then by taking → 0. Furthermore, if M is symmetrically right-continuous with respect to the distance at To prove Part (ii), using Part (ii) of Proposition 1 and (2) in the second observation of Remark 1, we have From Part (ii) of Proposition 2, we also have To prove Part (iii), using Part (ii) of Proposition 1 and (3) in the third observation of Remark 1, we have From Part (ii) of Proposition 2, we also have To prove Part (iv), using Part (iii) of Proposition 1 and ( 4) and (8) in the fourth observation of Remark 1, we have and From Part (ii) of Proposition 2, we also have Therefore, we can similarly obtain the desired results, and the proof is complete.
Proposition 7. Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞).Assume that the following inequality is satisfied we have the following properties.
(i) Suppose that M satisfies the -triangle inequality, and that t n → t Then, the following statements hold true.

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If M is continuous with respect to the distance at t • , then • If M is symmetrically continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , } and that t n → t If M is continuous or symmetrically continuous with respect to the distance at t • , then (iii) Suppose that M satisfies the -triangle inequality and that t n → t is continuous or symmetrically continuous with respect to the distance at t • , then Proof.From Propositions 5 and 6, we have This completes the proof.
Proposition 8. Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞).Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality, that t n → t • , x n M −→ x • and y n M −→ y • as n → ∞, and that {M(x n , y n , t n )} ∞ n=1 is an increasing sequence.
• If M is right-continuous with respect to the distance at t • , then • If M is symmetrically right-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the -triangle inequality, that t n → t n=1 is an increasing sequence.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then n=1 is an increasing sequence.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then n=1 is an increasing sequence.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then Proof.To prove Part (i), given any fixed > 0, there exists n 0 ∈ N such that t n − t • < for all n ≥ n 0 .Using Part (i) of Proposition 1 and the first observation of Remark 1, we have for all n ≥ n 0 .To prove the first case of Part (i), since using Part (vi) of Lemma 2, we have If M is right-continuous with respect to the distance at t • , then by taking → 0. Furthermore, if M is symmetrically right-continuous with respect to the distance at t • , then lim The remaining proof follows from the arguments of the proof of Proposition 6.This completes the proof.
Proposition 9. Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞).Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality, that t n → t n=1 is an increasing sequence.Then, the following statements hold true.
• If M is continuous with respect to the distance at t • , then • If M is symmetrically continuous with respect to the distance at t • , then n=1 is an increasing sequence.If M is continuous or symmetrically continuous with respect to the distance at t • , then (iii) Suppose that M satisfies the -triangle inequality, that t n → t n=1 is an increasing sequence.If M is continuous or symmetrically continuous with respect to the distance at t • , then Proof.From Propositions 5 and 8, we have This completes the proof.
From the triangle inequality of d, we have Therefore, we obtain Finally, we obtain Definition 6.Let (X, d) be a metric space, and let (X, M) be a fuzzy semi-metric space.
• Given any fixed y ∈ X and 0 < t < ∞, we consider the mapping M(•, y, t) : X → [0, 1].We say that the mapping M(•, y, t) is -continuous at x with respect to d if and only if x n d −→ x as n → ∞ implies M(x n , y, t) → M(x, y, t) as n → ∞.

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Given any fixed x ∈ X and 0 < t < ∞, we consider the mapping M(x, •, t) : X → [0, 1].We say that the mapping M(x, •, t) is -continuous at x with respect to d if and only if y n d −→ y as n → ∞ implies M(x, y n , t) → M(x, y, t) as n → ∞.Proposition 10.Let (X, d) be a metric space; let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component; and let {( Assume that the following conditions are satisfied
Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality.
• If M is left-continuous with respect to the distance at t • , then • If M is symmetrically left-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.If M is left-continuous or symmetrically left-continuous with respect to the distance at t • , then according to the assumption for continuities, we have which correspond to (9).The arguments in the proof of Proposition 5 are still valid, and the proof is complete.
Proposition 11.Let (X, d) be a metric space; let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component; and let for any sequences {a n } ∞ n=1 and {b n } ∞ n=1 in [0, 1] and that the following conditions are satisfied
Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality.
• If M is right-continuous with respect to the distance at t • , then • If M is symmetrically right-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then according to the assumption for continuities, we have which correspond to (11).The arguments in the proof of Proposition 6 are still valid, and the proof is complete.
Proposition 12. Let (X, d) be a metric space; let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component; and let {( and that the following conditions are satisfied is -continuous at y • with respect to d, and the mapping M(y Given t • > 0, we have the following properties. (i) Suppose that M satisfies the -triangle inequality.
• If M is continuous with respect to the distance at t • , then • If M is symmetrically continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.If M is continuous or symmetrically continuous with respect to the distance at t • , then Proof.From Propositions 10 and 11, we have This completes the proof.
Proposition 13.Let (X, d) be a metric space, and let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component.
n=1 is an increasing sequence.Assume that the following conditions are satisfied

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given any fixed t ∈ (0, ∞), the mapping M(x Given t • > 0, we have the following properties. (i) Suppose that M satisfies the -triangle inequality.
• If M is right-continuous with respect to the distance at t • , then • If M is symmetrically right-continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then according to the assumption for continuities, we have which correspond to (11).The arguments in the proof of Proposition 8 are still valid, and the proof is complete.
Proposition 14.Let (X, d) be a metric space; let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous with respect to the first or second component; and let n=1 is an increasing sequence.Assume that the following conditions are satisfied
Given t • > 0, we have the following properties.
(i) Suppose that M satisfies the -triangle inequality.
• If M is continuous with respect to the distance at t • , then • If M is symmetrically continuous with respect to the distance at t • , then (ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , , }.If M is continuous or symmetrically continuous with respect to the distance at t • , then Proof.From Propositions 10 and 13, we have This completes the proof.

Dual Metric Convergence Based on the Infimum
Recall that the fuzzy semi-metric M is a mapping from X × X × [0, ∞) into [0, 1].Now, we are going to consider the dual sense by considering the mapping from (0, 1] × X × X into [0, ∞), which will be named as a dual fuzzy semi-metric.The potential application of dual fuzzy semi-metric will be studying the fixed point theorems in fuzzy semi-metric space by referring to Wu [8].The dual fuzzy semi-metric was called as the auxiliary function in Wu [8] for only considering the -triangle inequality.In this paper, we shall extend to study the dual fuzzy semi-metric by considering the •-triangle inequality for • ∈ { , , , }.
For any fixed x, y ∈ X, Proposition 1 says that the mapping M(x, y, •) is nondecreasing, where the value M(x, y, t) is interpreted as the membership degree of the distance between x and y that is less than t.Therefore, when t is sufficiently large, it is reasonable to argue that the membership degree M(x, y, t) will be close to one.Alternatively, when t is sufficiently small, the membership degree M(x, y, t) will be close to zero.Therefore, we propose the following definition.Definition 7. Let (X, M) be a fuzzy semi-metric space.

•
We say that M satisfies the canonical condition if and only if lim t→+∞ M(x, y, t) = 1 for any fixed x, y ∈ X. = 0 for x = y.
Therefore, M satisfies both the canonical and rational conditions.
The concept of dual fuzzy semi-metric is defined below.Definition 8. Let (X, M) be a fuzzy semi-metric space such that M satisfies the canonical condition.Given any fixed x, y ∈ X and any fixed λ ∈ (0, 1], we define the set The mapping Γ ↓ from (0, 1] × X × X into [0, ∞) is also called the infimum type of dual fuzzy semi-metric.
Remark 4. We have the following observations.
In the sequel, we are going to establish the triangle inequalities and the convergence for the infimum type of dual fuzzy semi-metric Γ ↓ .Therefore, we shall first provide some relationships between the fuzzy semi-metric M and the infimum type of dual fuzzy semi-metric Γ ↓ .Proposition 15.Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the canonical and rational conditions.Suppose that M satisfies the •-triangle inequality for • ∈ { , , , }.Then, given any fixed x, y ∈ X with x = y, we have Γ ↓ (λ, x, y) = 0, i.e., Γ ↓ (λ, x, y) > 0 for λ ∈ (0, 1).
We consider the following cases.
• Suppose that M satisfies the -triangle inequality.By Part (iii) of Proposition 1, we have Since M satisfies the rational condition, it follows that Therefore, we conclude that Γ ↓ (λ, x, y) = 0 for λ ∈ (0, 1).This completes the proof.
Proposition 16.Let (X, M) be a fuzzy semi-metric space such that M satisfies the canonical condition.Given any fixed x, y ∈ X and any fixed λ ∈ (0, 1), we have the following properties.
(i) If > 0 is sufficiently small such that < Γ ↓ (λ, x, y), then If we further assume that M is left-continuous with respect to the distance on (0, ∞) and that (ii) For any > 0, we have the following properties.
To prove Part (ii), we consider the following cases.
In order to establish the triangle inequalities for the infimum type of dual fuzzy semi-metric, we need to provide a useful lemma.Lemma 3. Suppose that the t-norm * is left-continuous at one with respect to the first or second component.
Proof.If r n ↑ 1, then, using Part (ii) of Proposition 1, we have r n * r n ↑ 1.Therefore, for any 0 < a < 1, there exists r ∈ (0, 1) such that a < r * r.In general, using the increasing property of the t-norm, we have According to the the increasing property of the t-norm, we obtain This completes the proof.
Theorem 1. (Triangle inequalities for the dual fuzzy semi-metric) Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the canonical condition and the t-norm * is left-continuous at one in the first or second component.
It is clear to see that x n M −→ x if and only if Γ ↓ (λ, x n , x) → 0, and x n M −→ x if and only if Γ ↓ (λ, x, x n ) → 0 for all λ ∈ (0, 1).
Next, we consider the Cauchy sequence that will be useful for studying the fixed point theorems in fuzzy semi-metric space.Definition 9. Let (X, M) be a fuzzy semi-metric space, and let {x n } ∞ n=1 be a sequence in X.

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

•
We say that {x n } ∞ n=1 is a Cauchy sequence in the metric sense if and only if, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N such that 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 10.Let (X, M) be a fuzzy semi-metric space such that M satisfies the canonical condition, and let {x n } ∞ n=1 be a sequence in X.

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Given any fixed λ ∈ (0, 1), we say that {x n } ∞ n=1 is a λ-Cauchy sequence with respect to Γ ↓ if and only if, given any > 0, there exists n ∈ N such that m, n ≥ n implies Γ ↓ (λ, x m , x n ) < and Γ ↓ (λ, x n , x m ) < .Theorem 3. Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the canonical condition and the t-norm * is left-continuous at one with respect to the first or second component.Suppose that M satisfies the •-triangle inequality for • ∈ { , , , }.Let {x n } ∞ n=1 be a sequence in X.Then, we have the following properties.
(i) {x n } ∞ n=1 is a >-Cauchy sequence in the metric sense if and only if it is a (λ, >)-Cauchy sequence with respect to Γ ↓ for all λ ∈ (0, 1).(ii) {x n } ∞ n=1 is a <-Cauchy sequence in the metric sense if and only if it is a (λ, <)-Cauchy sequence with respect to Γ ↓ for all λ ∈ (0, 1).
Proof.It suffices to prove Part (i).Suppose that {x n } ∞ n=1 is a >-Cauchy sequence in the metric sense, which says that, 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.In other words, for any fixed λ ∈ (0, 1), given any > 0, there exists n ∈ N such that m > n ≥ n implies M(x m , x n , /2) > 1 − λ.By the definition of Γ λ , we obtain for m > n ≥ n , which shows that {x n } ∞ n=1 is a (λ, >)-Cauchy sequence with respect to Γ ↓ .For the converse, given any δ > 0 and λ ∈ (0, 1], there exists n δ,λ ∈ N such that m > n ≥ n δ,λ implies Γ ↓ (λ, x m , x n ) < δ, which also says that, for any fixed t > 0, given any ∈ (0, 1), there exists n ∈ N such that m > n ≥ n implies Γ ↓ ( /2, x m , x n ) < t.Therefore, we obtain for m > n ≥ n by Part (i) of Proposition 17.This shows that {x n } ∞ n=1 is a >-Cauchy sequence in the metric sense, and the proof is complete.
It is clear to see that {x n } ∞ n=1 is a >-Cauchy sequence in the metric sense if and only if it is a (λ, >)-Cauchy sequence with respect to Γ ↓ for all λ ∈ (0, 1).

Dual Metric Convergence Based on the Supremum
Recall that the infimum type of dual fuzzy semi-metric is based on the infimum in which the canonical conditions are assumed to be satisfied.The purpose for considering the canonical condition is to guarantee the infimum type of dual fuzzy semi-metric space is well-defined.
To prove Part (i), using Parts (i) and (ii) of Proposition 18, we have t ≤ Γ ↓ (λ, x, y), which says that Γ ↑ (λ, x, y) − < Γ ↓ (λ, x, y).Since can be any positive real number, we obtain the desired inequality.Part (ii) can be similarly obtained by considering t ≤ Γ ↓ (λ, y, x).This completes the proof.Proposition 23.Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the rational condition.Given any fixed x, y ∈ X with x = y and any fixed λ ∈ (0, 1), we have the following properties.
To prove Part (ii), we consider the following cases.
This completes the proof.
Proposition 24.Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that M satisfies the rational condition.Given any fixed x, y ∈ X with x = y and any fixed λ ∈ (0, 1), we have the following properties.

Example 2 .
y, then x = y.• If x n M −→ x and x n M −→ y, then x = y.(ii) Suppose that M satisfies the -triangle inequality.If x n M −→ x and x n M −→ y, then x = y.In other words, the M -limit is unique.(iii) Suppose that M satisfies the -triangle inequality.If x n M −→ x and x n M −→ y, then x = y.In other words, the M -limit is unique.Continued from Example 1, we see that x n M −→ x as n → ∞ if and only if (i) Given any fixed a, b ∈ (0, 1], suppose that the t-norm * is left-continuous at a and b with respect to the first or second component.If {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] such that a n → a− and b n → b− as n → ∞, then a n * b n → a * b as n → ∞. (ii) Given any fixed a, b ∈ [0, 1), suppose that the t-norm * is right-continuous at a and b with respect to the first or second component.If {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] such that a n → a+ and b n → b+ as n → ∞, then a n * b n → a * b as n → ∞.

1 Using
Part (i), we obtain sup n

(
iv) Suppose that M satisfies the -triangle inequality, and that t n → t • , x n M −→ x • and y n M −→ y • as n → ∞ simultaneously, or x n M −→ x • and y n M −→ y • as n → ∞ simultaneously.If M is right-continuous or symmetrically right-continuous with respect to the distance at t • , then

Remark 3 .
The convergence of this example does not need any extra sufficient conditions.If the inequality (10) is satisfied, then, by Part (i) of Lemma 2, sup n (a n * b n ) = sup n a n * sup n b n According to Part (ii) of Lemma 2, the inequality (10) can be replaced by the following statement: given any sequences {a n } ∞ n=1 and {b n } ∞ n=1 , for any m, n ∈ N, there exists n 0 ∈ N such that a m * b n ≤ a n 0 * b n 0 .

Example 6 .
Continued from Examples 2 and 5, we have x n M −→ x if and only if lim n→∞ d(x n , x) = 0 x n M −→ x if and only if lim n→∞

Example 7 .
Continued from Examples 1 and 5, we haveM(x m , x n , t) > 1 − r if and only if t t + d(x m , x n ) > 1 − r if and only if 1 λ − 1 • d(x m , x n ) < t and Γ ↓ (λ, x m , x n ) < if and only if 1