# The Decomposition Theorem for a Fuzzy Quasinorm

Authors: Rui Gao; Xinxin Li; Jianrong Wu
Journal: Journal of Mathematics
 (2020)

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## 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 generalt-norm, characterize fuzzy quasinorms in terms of families of star quasiseminorms, and establish the connection between the fuzzy quasinorm and the family of quasinorms.


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## Body

## 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 X×X×−∞,∞ 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 X, a fuzzy norm can be determined on X. 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 N with general t-norm into the associated family of star quasiseminorms, denoted by PN. Additionally, we prove that an increasing and separating family of star quasiseminorms P induces a fuzzy quasinorm, denoted by NP. Moreover, we show that a family of star quasiseminorms coincides with the family of star quasiseminorms associated to the fuzzy quasinorm induced by P (see Theorem 4); meanwhile, a fuzzy quasinorm N coincides with the fuzzy quasinorm induced by the family of star quasiseminorms associated to N (see Theorem 5). That is, P=PNP and N=NPN. 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. LetZ+ be the set of all positive integers, R=−∞,∞, R+=0,∞.First, let us recall the concept of continuoust-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):a∗1=a, ∀a∈0,1(T4):a∗b≤c∗d whenever a≤c and b≤d and a,b,c,d∈0,1
The following are examples of some t-norms that are frequently used as fuzzy intersections deﬁned for alla,b∈0,1.(i)
Standard intersection:aΔ1b = a ∧ b(ii)
Algebraic product:aΔ2b=ab(iii)
Bounded difference:aΔ3b=maxa+b−1,0Definition 2.
(see [10]). A fuzzy quasinorm on a real linear space X is a pair N,∗, or N for simplicity, such that ∗ is a continuous t-norm and N is a fuzzy set in X×0,∞ satisfying the following conditions for every x,y∈X:(i)
(FQN1):Nx,0=0(ii)
(FQN2):Nx,t=N−x,t=1 for all t>0 if and only if x=0(iii)
(FQN3):Ncx,t=Nx,t/c for all c∈R+(iv)
(FQN4):Nx,t∗Ny,s≤Nx+y,t+s for all s,t≥0(v)
(FQN5):Nx,⋅: 0,∞⟶0,1 is left continuous(vi)
(FQN6):limt⟶∞Nx,t=1Remark 1.
We point out that (FQN2) and (FQN4) imply thatNx,⋅ is increasing.
The following condition will be used in the paper:
(FQN7): forx≠0, Nx,⋅ is strictly increasing on t:0<Nx,t<1.
A fuzzy norm [1] on a real linear space X is a fuzzy quasinorm N,∗ on X such that Ncx,t=Nx,t/c for all x∈X and c∈R/0. 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 tripleX,N,∗ such that X is a real linear space and N,∗ is a fuzzy (quasi-) norm on X.
Each fuzzy quasinormN,∗ on X defines a topology (denoted by τN) on X by taking all open balls as a base:(1)BNx,r,t:x∈X,r∈0,1,t>0,where BNx,r,t = y∈X:Ny−x,t>1−r. BNx,r,t is said to be an open ball with center x and radius r. It is easy to see that the topology τN is T0.
In the rest of the paper, the notationαn↑α (αn↓α, resp.) means that an increasing (decreasing, resp.) sequence of real numbers αn convergences to a real number α.


## 3. Quasiseminorm Structures in a Fuzzy Quasinormed Space

It is well known that a quasinorm on a real linear spaceX is a function p: X⟶[0, ∞) satisfying the conditions: for all x,y∈X and c∈R+,(i)
(QN1):px=p−x=0⇒x=0(ii)
(QN2):pcx=cpx(iii)
(QN3):px+y≤px+pyIfp satisﬁes only the conditions (QN2) and (QN3), then it is called a quasiseminorm.Remark 2.
From (QN2), we getp0 = 0.Proposition 1.
LetX,N,∗ be a fuzzy quasinormed space, and let α∈0,1. The function ⋅α: X⟶0,∞ is given by(2)xα=inft>0:Nx,t≥α.Then, for allx∈X and t∈R+,(1)
xα is increasing with respect to α∈0,1(2)
xα = supt>0:Nx,t<α(3)
Nx,t≥α implies that xα≤t, and equivalently, xα>t implies that Nx,t<α(4)
Nx,t>α implies that xα<t, and xα<t implies that Nx,t≥αThe proof is direct and omitted.⋅α:α∈0,1 is called the family of “α-quasiseminorms” corresponding to fuzzy quasinorm N and denoted by PN.Proposition 2.
LetX,N,∗ be a fuzzy quasinormed space, and let x∈X. If Nx,⋅ is continuous and satisfies (FQN7), then xα is strictly increasing with respect to α∈0,1.Proof.
Letα1,α2∈0,1 with α1<α2. Since Nx,⋅ is strictly increasing and continuous, there exist t1,t2∈R+ such that Nx,t1=α1, Nx,t2=α2 and Nx,t∈α1,α2 when t∈t1,t2. It is easy to see that xα2=t2>t1=xα1. Thus, xα is strictly increasing with respect to α∈0,1.Proposition 3.
LetX,N,∗ be a fuzzy quasinormed space, and let x∈X. Then, the following assertions are equivalent:(1)
Nx,⋅ satisfies (FQN7)(2)
xα = inft>0:Nx,t>α=supt>0:Nx,t≤α for each α∈0,1(3)
Nx,t>α if and only if xα<t, that is, Nx,t≤α if and only if xα≥tProof.
(i)
(1) ⇒ (3): from Proposition1 (4), we only have to show xα<t implies Nx,t>α. In fact, if xα<t, by the definition of xα, there is 0<t1<t such that Nx,t1≥α. Thus Nx,t>α whenever Nx,⋅ satisfies (FQN7).(ii)
(3) ⇒ (2): let(3)t0=inft>0:Nx,t>α.In the light of (2), we get xα≤t0. If xα<t0, we have Nx,t0>α from (3). Since Nx,⋅ is left continuous, there is δ>0 such that Nx,t0−δ>α, which conflicts with the definition of t0. Thus,(4)xα=t0=inft>0:Nx,t>αinft>0:Nx,t>α=supt>0:Nx,t≤α.can be shown by using the similar technique used in Proposition1 (2).(iii)
(2)⇒(1): suppose thatNx,⋅ is not strictly increasing. Then, there exist t1,t2∈t:0<Nx,t<1 such that t1<t2 and Nx,⋅≡α0 on t1,t2. Thus,(5)supt>0:Nx,t≤α0≥t2>t1≥inft>0:Nx,t≥α0=xα0,which conflicts with the supposition (2).Definition 3.
LetI⊆R. A family of real-valued maps di:i∈I will be called(1)
lower semicontinuous (shortly LSC) if for anyi∈I, di = infj>idj(2)
upper semicontinuous (shortly USC) if for anyi∈I, di = supj<idj(3)
continuous if it is both LSC and USC
The following lemma is obvious.Lemma 1.
LetI be an interval in R. An increasing family of real-valued maps di:i∈I is(1)
LSC if and only for any sequenceαn in I with αn↓α∈I, dα = limn⟶∞dαn(2)
USC if and only for any sequenceαn in I with αn↑α∈I, dα = limn⟶∞dαnProposition 4.
LetX,N,∗ be a fuzzy quasinormed space, and let α∈0,1, t>0. xα is defined by (2). Then,(1)
⋅α:α∈0,1 is USC(2)
⋅α:α∈0,1 is LSC if and only if Nx,⋅ satisfies (FQN7)Proof.
(1)
Letαn,α∈0,1 and αn↑α. From Proposition 1 (1), xαn is increasing and limn⟶∞xαn≤xα. If limn⟶∞xαn<xα, then there is a t0>0 such that limn⟶∞xαn<t0<xα, and hence, xαn<t0 for all n∈Z+, which together with Proposition 1 (4) follows that Nx,t0≥αn for all n∈Z+. Thus, Nx,t0≥α. Hence, xα≤t0. This is a contradiction. Thus, limn⟶∞xαn=xα.(2)
SupposeNx,⋅ satisfies (FQN7). Let αn,α∈0,1 and αn↑α; then, limn⟶∞xαn≥xα. Suppose limn⟶∞xαn>xα. Then, there is a t0>0 such that limn⟶∞xαn>t0>xα, and then, xαn>t0 for all n∈Z+. By Proposition 1 (3), we have Nx,t0<αn for all n∈Z+. Therefore, Nx,t0≤α.Case 1:Nx,t0<α. From Proposition 1 (2), we get xα>t0. This is a contradiction.Case 2:Nx,t0=α. Since Nx,⋅ satisfies (FQN7), Nx,t0−ε<α for any ε>0. Then, xα>t0−ε from Proposition 1 (3). Hence, xα≥t0. This is a contradiction.
Combining the abovementioned discussion, we knowlimn⟶∞xαn=xα. From Lemma 1, we know that ⋅α:α∈0,1 is LSC.
Now, we suppose⋅α:α∈0,1 is LSC. Let t1,t2∈R+ with t1<t2. Then, Nx,t1≤Nx,t2. Suppose that Nx,t1=Nx,t2; then, Nx,⋅≡α0 on t1,t2. Let m=supt:Nx,t=α0; then,(6)xα0=inft>0:Nx,t≥α0≤t1<t2≤m.
SinceNx,⋅ is left continuous, Nx,m=α0. For any α′>α0 and any t>0 with Nx,t≥α′, we have t>m; hence, xα′ = inft>0:Nx,t≥α′≥m. Let αn⊆0,1 be a strictly decreasing sequence with αn↓α0; then, limn⟶∞xαn≥m>xα0, which conflicts with the supposition. Thus, Nx,t1<Nx,t2. That is, Nx,⋅ satisfies (FQN7).Definition 4.
LetX be a linear space and ∗ be a continuous t-norm. For each α∈0,1, pα is a function from X to R+. P=pα:α∈0,1 is called a family of star quasiseminorms if it satisfies the conditions: for all x,y∈X, α,β∈0,1, and c∈0,∞,(∗QN1):pαcx=cpαx(∗QN2):pα∗βx+y≤pαx+pβy
IfP satisfies the condition (∗QN3): pαx = 0 for every α∈0,1 implies x = 0, then P is said to be separating.Remark 3.
From (∗QN1), we knowpα0 = 0 for every α∈0,1.Remark 4.
If ∗ = ∧, then a family of star quasiseminorms is just a family of quasiseminorms.
The following result is obvious:Proposition 5.
LetP=pα:α∈0,1 be a family of star quasiseminorms. For each x∈X, let(7)UPx=UPx;α1,α2,…,αn;ε:ε>0;α1,α2,…,αn∈0,1,n∈Z+,where(8)UPx;α1,α2,…,αn;ε=y∈X:pαiy−x<ε,αi∈0,1,i=1,2,…,n∩i=1ny∈X:pαiy−x<ε,αi∈0,1y∈X:pmaxαi:1≤i≤ny−x<ε.Then,UPx is a basis of neighborhoods of x.The topology takingUPx as a basis of neighborhoods of x is said to be the topology induced by P and denoted by τP. It is easy to show that τP is T0 if P is separating.Theorem 1.
LetX,N,∗ be a fuzzy quasinormed space. PN=⋅α:α∈0,1 where ⋅α is defined by (2) for all α∈0,1. Then,(1)
PN is a separating family of ∗ quasiseminorms(2)
the topologyτPN induced by PN coincides the topology τNProof.
(1)
Letx,y∈X, α∈0,1, and c∈R+.(∗QN1):cxα=inft>0:Ncx,t≥α=inft>0:Nx,t/c≥α=infct′>0:Nx,t′≥α=cxα.(∗QN2): for anyt1>xα and t2>yβ, from the definition of ⋅α, there exist t1∗,t2∗>0 such that t1∗<t1, t2∗<t2, Nx,t1∗≥α and Ny,t2∗≥β. Hence,(9)Nx+y,t1∗+t2∗≥Nx,t1∗∗Ny,t2∗≥α∗β.Therefore,(10)x+yα∗β=inft>0:Nx+y,t≥α∗β≤t1∗+t2∗<t1+t2.By the arbitrariness oft1 and t2, we know that x+yα∗β≤xα+yβ.(∗QN3): ifxα = 0 for every α∈0,1, then xα<t for all t>0; hence, Nx,t≥α from Proposition 1 (4). By the arbitrariness of α∈0,1, we get Nx,t=1 for all t>0. In light of (FQN2), we have x=0.(2)
For anyx,y∈X, r∈0,1 and t>0, if Ny−x,t>1−r, then y−x1−r<t from Proposition 1 (4). Therefore, BNx,r,t⊆UPNx;1−r;t.On the other hand, for anyx,y∈X, α∈0,1, and t>0, if y−xα<t, then there is 0<t1<t such that Ny−x,t1≥α from (2). Hence,(11)Ny−x,t≥Ny−x,t1>1−r,for any1>r>1−α. So, UPNx;α;t⊆BNx,r,t.Combining the abovementioned discussion, we getτN = τPN.Remark 5.
PN=⋅α:α∈0,1 is said to be the family of star quasiseminorms associated with the fuzzy quasinorm N,∗.Example 1.
LetX,⋅ be a quasinormed space, and let N: X×0,∞⟶[0, 1] given by(12)Nx,t=0,t=0,tt+x,t>0,for all x∈X. It is well known that N,∗ is a fuzzy quasinorm on X, where ∗ is any continuous t-norm. This fuzzy quasinorm is called the standard fuzzy quasinorm induced by ⋅. Obviously, Nx,t≥α is equivalent to t≥α/1−αx for any α∈0,1. So, it follows from (2) that(13)xα=α1−αx.
Therefore,PN=α/1−α⋅:α∈0,1.Theorem 2.
LetX,N,∗ be a fuzzy quasinormed space. If for any x∈X, there exists t>0 such that Nx,t=1. The function ⋅1: X⟶[0, ∞) is given by(14)x1=inft>0:Nx,t=1,then, ⋅1 is a quasinorm on X. Moreover,(15)x1=supα∈0,1xα=limα⟶1−0xα.Proof.
(i)
(QN1): supposex1=−x1 = 0. From (2), for any t>0, there exists 0<t′,t″<t such that Nx,t′=1 and N−x,t″=1; therefore, Nx,t=N−x,t = 1, which together with (FQN2) implies that x=0.(ii)
(QN2): letλ>0. Then,(16)λx1=inft>0:Nλx,t=1=inft>0:Nx,tλ=1=infλt′>0:Nx,t′=1=λx1.(iii)
(QN3): take anyt1>x1 and t2>y1. From (2), there exist t1∗,t2∗>0 such that t1∗<t1, t2∗<t2, Nx,t1∗=1 and Ny,t2∗=1. Thus,(17)Nx+y,t1∗+t2∗≥Nx,t1∗∗Ny,t2∗=1∗1=1.
Therefore,(18)x+y1=inft>0:Nx+y,t=1≤t1∗+t2∗.
By the arbitrariness oft1 and t2, we know that x+y1≤x1+y1. Thus, ⋅1 is a quasinorm.
Now, we prove (15). Since xα:α∈0,1 is increasing, it is easy to see that(19)x1≥supα∈0,1xα=limα⟶1−0xα,for all x∈X. For any x∈X, from the definition of x1, we know t>x1−ε when Nx,t=1. Thus, Nx,⋅<1 on 0,x1−ε. Hence, Nx,x1−ε<1. Take α0>0 such that Nx,x1−ε<1−α0. By Proposition 1 (4), we get x1−α0≥x1−ε. Consequently,(20)x1≤supα∈0,1xα.
The inequalities (19) and (20) imply equation (15).Theorem 3.
LetP=⋅α:α∈0,1 be an increasing separating USC family of star quasiseminorms on a real linear space X. For all x∈X and t>0, let X×0,∞⟶0,1 be given by(21)NPx,t=0,t=0,supα∈0,1:xα<t,t>0.Then,(1)
the pairNP,∗ is a fuzzy quasinorm on X(2)
the topologyτNP induced by fuzzy quasinorm NP coincides the topology τP induced by PProof.
(1)
(FQN1) is obvious.(FQN2): ifNPx,t = 1 for all t>0, then xα<t for all α∈0,1 from (21). Therefore, xα = 0 for all α∈0,1. Since P is separating, x = 0. Conversely, if x = 0, then xα=0<t for all t>0 from Remark 3. Hence, NPx,t=NP−x,t=1 from (21).(FQN3): letc∈R+. From (∗QN1), we have(22)NPcx,t=supα∈0,1:cxα<t=supα∈0,1:xα<tc=NPx,t/c.(FQN4):letx,y∈X and s,t>0. Set NPx,t=β, NPy,s=γ. For any ε>0 with ε<minβ,γ, there exist α′,α″∈0,1 such that α′>β−ε, α″>γ−ε, xα′<t, and yα″<s. Therefore, xβ−ε<t and yγ−ε<s. Hence,(23)x+yβ−ε∗γ−ε≤xβ−ε+yγ−ε<t+s.It follows from (21) that(24)NPx+y,t+s≥β−ε∗γ−ε.By the arbitrariness ofε>0 and the continuity of ∗, we know that(25)NPx+y,t+s≥β∗γ=NPx,t∗NPy,s.(FQN5): it is easy to see thatNP0,⋅≡1, and hence, it is continuous. Now, take x0∈X/0 and t0>0 arbitrarily. If NPx,t0=0, then NPx,t=NPx,t0=0 for all t<t0. So, NPx,⋅ is left continuous at t0. On the other hand, if 0<NPx,t0≤1. Given ε>0 arbitrarily, from (21), there exists α0∈(0, 1) such that xα0<t0 and NPx,t0−ε<α0. For any t with xα0<t<t0, we have NPx,t≥α0 by (21). Hence, NPx,t0−NPx,t≤NPx,t0−α0<ε. Therefore, NPx,⋅ is left continuous at t0.(FQN6): letx∈X and let ε>0. There exists α0∈0,1 such that 1−α0<ε. For any t>xα0, we have NPx,t≥α0>1−ε. Therefore, limt⟶∞NPx,t=1.(2)
For allx,y∈X, r∈0,1, and t>0, if NPy−x,t>1−r, then there is α∈1−r,1 such that y−xα<t; therefore, BNPx,r,t⊆UPx;α;t.
On the other hand, for allx,y∈X, α∈0,1, and t>0, if y−xα<t, then NPy−x,t≥α from the definition of NP. Hence, NPy−x,t>1−r for any 1>r>1−α. Therefore, UPx;α;t⊆BNPx,r,t.
Combining the abovementioned discussion, we getτNP = τP.Remark 6.
The abovementioned fuzzy quasinormNP,∗ is said to be induced by the family of star quasiseminorms P.Theorem 4.
LetP=⋅α:α∈0,1, NP,∗ be as in Theorem 3.3. PNP=⋅α:α∈0,1 is the family of star quasiseminorms associated with NP,∗. Then, xα = xα for all x∈X and α∈0,1. That is, PNP=P.Proof.
For anyt>xα, we know NPx,t≥α from (21), which together with Proposition 1 (3) implies that xα≤t. By the arbitrariness of t, we have xα≤xα.
For anyt>xα, from (2), we know there exists t0∈0,t such that NPx,t0≥α.(i)
Case 1:NPx,t0>α. From (21), there exists α0∈α,1 such that xα0<t0. Since xα:α∈0,1 is increasing, we have xα<xα0<t0<t. By the arbitrariness of t, we have xα≤xα.(ii)
Case 2:NPx,t0=α. From (21), there exists a strictly increasing sequence αn⊆0,α with αn↑α such that xαn<t0. Noting that P=⋅α:α∈0,1 is USC, we have xα≤t0<t. By the arbitrariness of t, we have xα≤xα.
Combining the abovementioned discussion, we getxα=xα.Theorem 5.
LetX,N,∗ be a fuzzy quasinormed space. Then, NPN=N.Proof.
LetPN=⋅α:α∈0,1 be the family of star quasiseminorms induced by the fuzzy quasinorm N, and let x∈X.
It is obvious thatNPNx,0 = 0 = Nx,0.
Lett>0. From (21) and Proposition 1 (4), we obtain that NPNx,t≤Nx,t directly. Now, we are going to show that NPNx,t≥Nx,t. Without loss of generality, we suppose that Nx,t>0. We take a strictly increasing sequence αn⊆0,Nx,t with αn↑Nx,t. From Proposition 1 (4), we get xαn<t. So, NPNx,t≥αn. Therefore, NPNx,t≥Nx,t.
Combining the abovementioned discussion, we getNPNx,t=Nx,t for all x∈X and t≥0, and hence, NPN=N.


## 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 generalt-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 [14–16]) and Pythagorean fuzzy set theory [17].



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