Abstract

This paper deals with fuzzy quasinormed spaces in the sense of Alegre and Romaguera. After introducing the concept of the family of star quasiseminorms, we prove the decomposition theorem for a fuzzy quasinorm with general t-norm, characterize fuzzy quasinorms in terms of families of star quasiseminorms, and establish the connection between the fuzzy quasinorm and the family of quasinorms.

1. Introduction

In 1984, Katsaras [1] first introduced an idea of fuzzy norm on a linear space. In 1992, Felbin [2] introduced the concept of fuzzy norm on a linear space whose associated metric is Kaleva and Seikkala type [3]. Inspired by the notion of probabilistic metric spaces, Kramosil and Michalek [4], in 1975, introduced the notion of fuzzy metric, a fuzzy set in the Cartesian product satisfying certain conditions. In 1994, Cheng and Mordeson [5] introduced an idea of fuzzy norm on a linear space in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type. Following Cheng and Mordeson, in 2003, Bag and Samanta [6] introduced a definition of fuzzy norm and proved the decomposition theorem of fuzzy norm to a family of crisp norms. This concept has been used in developing fuzzy functional analysis and its applications. Bag and Samanta [7] first considered a general t-norm in the definition of fuzzy normed linear space which was introduced in [6]and proved that if t-norm is chosen other than “min,” then the decomposition theorem of fuzzy norm may not hold. Since the decomposition theorem plays an important role in developing fuzzy functional analysis, it is worthy to establish a new kind of decomposition theorem of fuzzy norm with general t-norm. That is one of the goals of this article.

On the other hand, Bag and Samanta [6] stated that given an ascent family of norms on a real linear space , a fuzzy norm can be determined on . Furthermore, in 2009, Sadeqi and Kia [8] proved that a separating family of seminorms introduces a fuzzy norm in general, but it is not true in classical analysis. In 2014, Alegre and Romaguera [9] also dealt with fuzzy normed spaces in the sense of Cheng and Mordeson and characterized fuzzy norms in terms of ascending and separating families of seminorms. It is worth noting that the fuzzy norms mentioned above require a strong restriction on the particular choice of “min” for t-norm. Therefore, a natural query arises: how far the results of fuzzy normed linear spaces can be established with the fuzzy norm in its general form, i.e., waiving the restricted “min” t-norm in the triangle inequality. To deal with this problem is another goal of this article.

With the exception of symmetry of a fuzzy norm in [6], Alegre and Romaguera [10] introduced the concept of fuzzy quasinorm. They proved some results, such as the uniform boundedness theorem, in fuzzy quasinormed spaces in [11].

In this paper, we are going to conduct research in the framework of fuzzy quasinormed linear space introduced in [10]. After investigating some properties of “α-quasiseminorms” corresponding to a fuzzy quasinorm, we introduce a concept of a family of star quasiseminorms in Section 3. The novelty of this definition is the validity of a decomposition theorem for a fuzzy quasinorm with general t-norm into the associated family of star quasiseminorms, denoted by . Additionally, we prove that an increasing and separating family of star quasiseminorms induces a fuzzy quasinorm, denoted by . Moreover, we show that a family of star quasiseminorms coincides with the family of star quasiseminorms associated to the fuzzy quasinorm induced by (see Theorem 4); meanwhile, a fuzzy quasinorm coincides with the fuzzy quasinorm induced by the family of star quasiseminorms associated to (see Theorem 5). That is, and . Combining Theorems 4 and 5, we get the decomposition theorem for a fuzzy quasinorm with general t-norm, characterize fuzzy quasinorms in terms of families of star quasiseminorms, and establish the connection between the fuzzy quasinorm and the family of quasinorms.

2. Preliminaries

Throughout this paper, the symbols ∧ and ∨ mean the operator “min” and “max,” respectively. Let be the set of all positive integers, , .

First, let us recall the concept of continuous t-norms [12].

Definition 1. A binary operation : [0, 1] × [0, 1]⟶[0, 1] is a continuous t-norm if it satisfies the following conditions:(T1): is associative and commutative(T2): is continuous(T3): , (T4): whenever and and The following are examples of some t-norms that are frequently used as fuzzy intersections defined for all .(i)Standard intersection:  =  ∧ (ii)Algebraic product: (iii)Bounded difference:

Definition 2. (see [10]). A fuzzy quasinorm on a real linear space is a pair , or for simplicity, such that is a continuous t-norm and is a fuzzy set in satisfying the following conditions for every :(i)(FQN1): (ii)(FQN2): for all if and only if (iii)(FQN3): for all (iv)(FQN4): for all (v)(FQN5): : is left continuous(vi)(FQN6):

Remark 1. We point out that (FQN2) and (FQN4) imply that is increasing.
The following condition will be used in the paper:
(FQN7): for , is strictly increasing on .
A fuzzy norm [1] on a real linear space is a fuzzy quasinorm on such that for all and . Recall that if in Definition 2, we put ∗ = ∧, then one has the notion of a fuzzy norm as given by Cheng and Morderson [5].
By a fuzzy (quasi-) normed space, we mean a triple such that is a real linear space and is a fuzzy (quasi-) norm on .
Each fuzzy quasinorm on defines a topology (denoted by ) on by taking all open balls as a base:where  = . is said to be an open ball with center and radius . It is easy to see that the topology is .
In the rest of the paper, the notation (, resp.) means that an increasing (decreasing, resp.) sequence of real numbers convergences to a real number .

3. Quasiseminorm Structures in a Fuzzy Quasinormed Space

It is well known that a quasinorm on a real linear space is a function : ⟶[0, ∞) satisfying the conditions: for all and ,(i)(QN1): (ii)(QN2): (iii)(QN3):

If satisfies only the conditions (QN2) and (QN3), then it is called a quasiseminorm.

Remark 2. From (QN2), we get  = 0.

Proposition 1. Let be a fuzzy quasinormed space, and let . The function : is given by

Then, for all and ,(1) is increasing with respect to (2) = (3) implies that , and equivalently, implies that (4) implies that , and implies that

The proof is direct and omitted. is called the family of “α-quasiseminorms” corresponding to fuzzy quasinorm and denoted by .

Proposition 2. Let be a fuzzy quasinormed space, and let . If is continuous and satisfies (FQN7), then is strictly increasing with respect to .

Proof. Let with . Since is strictly increasing and continuous, there exist such that , and when . It is easy to see that . Thus, is strictly increasing with respect to .

Proposition 3. Let be a fuzzy quasinormed space, and let . Then, the following assertions are equivalent:(1) satisfies (FQN7)(2) =  for each (3) if and only if , that is, if and only if

Proof. (i)(1) ⇒ (3): from Proposition 1 (4), we only have to show implies . In fact, if , by the definition of , there is such that . Thus whenever satisfies (FQN7).(ii)(3) ⇒ (2): letIn the light of (2), we get . If , we have from (3). Since is left continuous, there is such that , which conflicts with the definition of . Thus,can be shown by using the similar technique used in Proposition 1 (2).(iii)(2)⇒(1): suppose that is not strictly increasing. Then, there exist such that and on . Thus,which conflicts with the supposition (2).

Definition 3. Let . A family of real-valued maps will be called(1)lower semicontinuous (shortly LSC) if for any ,  = (2)upper semicontinuous (shortly USC) if for any ,  = (3)continuous if it is both LSC and USCThe following lemma is obvious.

Lemma 1. Let be an interval in . An increasing family of real-valued maps is(1)LSC if and only for any sequence in with ,  = (2)USC if and only for any sequence in with ,  = 

Proposition 4. Let be a fuzzy quasinormed space, and let , . is defined by (2). Then,(1) is USC(2) is LSC if and only if satisfies (FQN7)

Proof. (1)Let and . From Proposition 1 (1), is increasing and . If , then there is a such that , and hence, for all , which together with Proposition 1 (4) follows that for all . Thus, . Hence, . This is a contradiction. Thus, .(2)Suppose satisfies (FQN7). Let and ; then, . Suppose . Then, there is a such that , and then, for all . By Proposition 1 (3), we have for all . Therefore, .Case 1: . From Proposition 1 (2), we get . This is a contradiction.Case 2: . Since satisfies (FQN7), for any >0. Then, from Proposition 1 (3). Hence, . This is a contradiction.Combining the abovementioned discussion, we know . From Lemma 1, we know that is LSC.
Now, we suppose is LSC. Let with . Then, . Suppose that ; then, on . Let ; then,Since is left continuous, . For any and any with , we have ; hence,  = . Let be a strictly decreasing sequence with ; then, , which conflicts with the supposition. Thus, . That is, satisfies (FQN7).

Definition 4. Let be a linear space and be a continuous t-norm. For each , is a function from to . is called a family of star quasiseminorms if it satisfies the conditions: for all , , and ,(∗QN1): (∗QN2):  If satisfies the condition (∗QN3):  = 0 for every implies  = 0, then is said to be separating.

Remark 3. From (∗QN1), we know  = 0 for every .

Remark 4. If ∗ = ∧, then a family of star quasiseminorms is just a family of quasiseminorms.
The following result is obvious:

Proposition 5. Let be a family of star quasiseminorms. For each , letwhere

Then, is a basis of neighborhoods of .

The topology taking as a basis of neighborhoods of is said to be the topology induced by and denoted by . It is easy to show that is T0 if is separating.

Theorem 1. Let be a fuzzy quasinormed space. where is defined by (2) for all . Then,(1) is a separating family of ∗ quasiseminorms(2)the topology induced by coincides the topology

Proof. (1)Let , , and .(∗QN1): .(∗QN2): for any and , from the definition of , there exist such that , , and . Hence,Therefore,By the arbitrariness of and , we know that .(∗QN3): if  = 0 for every , then for all ; hence, from Proposition 1 (4). By the arbitrariness of , we get for all . In light of (FQN2), we have .(2)For any , and , if , then from Proposition 1 (4). Therefore, .On the other hand, for any , , and , if , then there is such that from (2). Hence,for any . So, .Combining the abovementioned discussion, we get  = .

Remark 5. is said to be the family of star quasiseminorms associated with the fuzzy quasinorm .

Example 1. Let be a quasinormed space, and let : ⟶[0, 1] given byfor all . It is well known that is a fuzzy quasinorm on , where ∗ is any continuous t-norm. This fuzzy quasinorm is called the standard fuzzy quasinorm induced by . Obviously, is equivalent to for any . So, it follows from (2) thatTherefore, .

Theorem 2. Let be a fuzzy quasinormed space. If for any , there exists such that . The function : ⟶[0, ∞) is given bythen, is a quasinorm on . Moreover,

Proof. (i)(QN1): suppose  = 0. From (2), for any , there exists such that and ; therefore,  = 1, which together with (FQN2) implies that .(ii)(QN2): let >0. Then,(iii)(QN3): take any and . From (2), there exist such that , , and . Thus, Therefore,By the arbitrariness of and , we know that . Thus, is a quasinorm.
Now, we prove (15). Since is increasing, it is easy to see thatfor all . For any , from the definition of , we know when . Thus, on . Hence, . Take such that . By Proposition 1 (4), we get . Consequently,The inequalities (19) and (20) imply equation (15).

Theorem 3. Let be an increasing separating USC family of star quasiseminorms on a real linear space . For all and >0, let be given by

Then,(1)the pair is a fuzzy quasinorm on (2)the topology induced by fuzzy quasinorm coincides the topology induced by

Proof. (1)(FQN1) is obvious.(FQN2): if  = 1 for all , then for all from (21). Therefore,  = 0 for all . Since is separating,  = 0. Conversely, if  = 0, then for all from Remark 3. Hence, from (21).(FQN3): let . From (∗QN1), we have(FQN4):let and . Set , . For any with , there exist such that , , , and . Therefore, and . Hence,It follows from (21) thatBy the arbitrariness of and the continuity of ∗, we know that(FQN5): it is easy to see that , and hence, it is continuous. Now, take and arbitrarily. If , then for all . So, is left continuous at . On the other hand, if . Given arbitrarily, from (21), there exists (0, 1) such that and . For any with , we have by (21). Hence, . Therefore, is left continuous at .(FQN6): let and let . There exists such that . For any , we have . Therefore, .(2)For all , , and , if , then there is such that ; therefore, .On the other hand, for all , , and , if , then from the definition of . Hence, for any . Therefore, .
Combining the abovementioned discussion, we get  = .

Remark 6. The abovementioned fuzzy quasinorm is said to be induced by the family of star quasiseminorms .

Theorem 4. Let , be as in Theorem 3.3. is the family of star quasiseminorms associated with . Then,  =  for all and . That is, .

Proof. For any , we know from (21), which together with Proposition 1 (3) implies that . By the arbitrariness of , we have .
For any , from (2), we know there exists such that .(i)Case 1: . From (21), there exists such that . Since is increasing, we have . By the arbitrariness of , we have .(ii)Case 2: . From (21), there exists a strictly increasing sequence with such that . Noting that is USC, we have . By the arbitrariness of , we have .Combining the abovementioned discussion, we get .

Theorem 5. Let be a fuzzy quasinormed space. Then, .

Proof. Let be the family of star quasiseminorms induced by the fuzzy quasinorm , and let .
It is obvious that  = 0 = .
Let >0. From (21) and Proposition 1 (4), we obtain that directly. Now, we are going to show that . Without loss of generality, we suppose that . We take a strictly increasing sequence with . From Proposition 1 (4), we get . So, . Therefore, .
Combining the abovementioned discussion, we get for all and , and hence, .

4. Conclusions

This paper introduces a concept of the family of star quasiseminorms. With this new concept, the decomposition theorem for a fuzzy quasinorm with general t-norm is established, and the quasiseminorm structures in a fuzzy quasinormed space are revealed. Based on these results, the connection between the fuzzy quasinorm and the quasinorm is established. The proposed method provides a powerful tool to study the fuzzy functional analysis. Many results about the fuzzy functional analysis may be obtained easily from the corresponding versions in the functional analysis with the help of the decomposition theorem for a fuzzy quasinorm. Conversely, some topics about the functional analysis can be investigated in the view of the fuzzy functional analysis. For example, the characterizations of those pseudotopological linear spaces [13] that are fuzzy quasinormable may be investigated deeply. Moreover, the decomposition technique proposed in this paper may be applied in other research fields such as fuzzy Lie algebras (see [1416]) and Pythagorean fuzzy set theory [17].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Jianrong Wu was responsible for conceptualization, methodology, funding acquisition, and review and editing. Rui Gao and Xinxin Li contributed to formal analysis, investigation, and original draft preparation. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors acknowledge the support of the National Natural Science Foundation of China under Grant no. 11371013. The authors are grateful to the referees for their valuable comments which led to the improvement of this paper.