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Induced I(L)-fuzzy topological vector spaces
Fuzzy Sets and Systems 121 (2001) 293–299
www.elsevier.com/locate/fss
Induced I (L)-fuzzy topological vector spaces Jin-Xuan Fang ∗ , Cong-Hua Yan Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu, 210097, People’s Republic of China Received 28 December 1998; received in revised form 7 January 2000; accepted 14 February 2000
Abstract In this paper we introduce and study the induced I (L)-fuzzy topological vector spaces. The following main result is proved: If (LX ; ) is an L-fuzzy topological vector space, then (I (L)X ; !I (L) ()) is an I (L)-fuzzy topological vector space, where I (L) is the fuzzy unit interval and !I (L) () is the induced I (L)-fuzzy topology of the L-fuzzy topology . Furthermore, we also prove that the induced I (L)-fuzzy topological vector spaces (I (L)X ; !I (L) ()) preserve the product and quotient space c 2001 Elsevier Science B.V. All rights reserved. as well. Keywords: Fuzzy unit interval; Induced I (L)-fuzzy topology; L-fuzzy topological vector spaces; Induced I (L)-fuzzy topological vector spaces
1. Introduction and preliminaries In [6], Wang introduced the concept of induced I (L)-fuzzy topological spaces by using the I (L)valued lower semicontinuous mappings [4]. Let (LX ; ) be an L-fuzzy topological space. !I (L) () denotes the set of all I (L)-valued lower semicontinuous mappings on X . Then (I (L) X ; !I (L) ()) is an I (L)-fuzzy topological space and is called the induced I (L)-fuzzy topological space. Recently, we introduced and studied L-fuzzy topological vector spaces (see [1,2,10]). In [10], we proved that (X; T ) is a topological vector space i= (LX ; !L (T )) is an L-fuzzy topological vector space, where !L (T ) is the Project supported by the National Natural Science Foundation of China. ∗ Corresponding author. E-mail address: [email protected] (J.-X. Fang).
set of all L-valued lower semicontinuous mappings on X . Thus, a natural question can be presented: If (LX ; ) is an L-fuzzy topological vector space, is (I (L) X ; !I (L) ()) an I (L)-fuzzy topological vector space? In this paper we shall give a positive answer to this question. We prove that if is an L-fuzzy topology on the vector space X , then (I (L) X ; !I (L) ()) is an I (L)-fuzzy topological vector space i= (LX ; ) is an L-fuzzy topological vector space. Furthermore, we also show that the induced I (L)-fuzzy topological vector space (I (L) X ; !I (L) ()) preserves the product and the quotient space. Throughout this paper, L will denote a fuzzy lattice, i.e. a completely distributive lattice [5] with orderreversing involution → . 0 and 1 are its smallest and greatest elements, respectively. M (L) denotes the set of nonzero union irreducible elements in L. The elements of M (L) are also called molecules [7] in L.
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 3 7 - 3
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J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
For a ∈ L; ∗ (a) denotes a standard minimal family [7] of a. Given a nonempty set X; LX denotes the collection of all L-fuzzy sets on X . X denotes an L-fuzzy set which takes the constant value on X . Let I denote the unit interval [0; 1]. The L-fuzzy unit interval I (L) is the set of all equivalence classes [ ], where : R → L is a monotone decreasing mapping satisfying (t) = 1 for t¡0 and (t) = 0 for t¿1, and ∈ [ ] i= (t−) = (t−) and (t+) = (t+), for all t ∈ I . The natural L-fuzzy topology on I (L) is generated from the subbase {Lt ; Rt | t ∈ I }, where Lt [ ] = (t−) , and Rt [ ] = (t+). A partial order on I (L) is naturally deJned by [ ]6[] i= (t−)6(t−) and (t+)6(t+), for all t ∈ I . Let [ ] ∈ I (L), we deJne [ ] = [ ], where ∈ LR is deJned by (t) = (1 − t) for all t ∈ R. To simplify notation, we shall identify equivalence classes with their representatives in the following. Let ∈ I (L), and t ∈ R, then we have (t−) = (t+) =
{ (s+) | s ¡ t};
Denition 1.3 (Kubiak [4] and Wang [6]). For each t ∈ I , deJne mappings t ; !t : I (L) X → LX as follows: t () = −1 (Rt ); for all ∈ I (L)X : Obviously, t ()(x) = (x)(t+);
Denition 1.1 (Fang and Yan [1]). An L-fuzzy topology on X is a subfamily with the following properties: (1) X ∈ for every ∈ L; (2) is closed under arbitrary unions; (3) is closed under Jnite intersections. The pair (LX ; ) is called an L-fuzzy topological space (for short, L-fts). We write = {A | A ∈ }. The member of is called open L-fuzzy set and the member of is called closed L-fuzzy set. Denition 1.2 (Wang [7]). Let (LX ; ) be an L-fts, x ∈ M ∗ (LX ). A ∈ LX is called an R-neighborhood of x if there exists a P ∈ such that x ∈= P and A ⊂ P. Obviously, if P ∈ and x ∈= P, then P is a closed R-neighborhood of x . The set of all closed R-neighborhoods of x is denoted by − (x ).
!t ()(x) = (x)(t−):
Denition 1.4 (Kubiak [4] and Wang [6]). Let (LX ; ) be an L-fts. A mapping : X → I (L) is called I (L)-valued lower (resp. upper) semicontinuous if for each t ∈ I; t () ∈ (resp. (!t ()) ∈ ). Denition 1.5 (Wang [6]). DeJne the mapping ∗ : LX → I (L) X satisfying P ∗ (x)(t+) 1; = P(x); 0;
{ (s−) | s ¿ t}:
From this it is easy to prove that the following statements are equivalent: (1) ; ∈ I (L); = ; (2) (t+) = (t+) for all t ∈ I ; (3) (t−) = (t−) for all t ∈ I .
!t () = −1 (Lt )
t¡0; 06t¡1; t¿1
(1.1)
for all P ∈ LX ; x ∈ X: Moreover, for each t ∈ R, deJne a constant mapping tX∗ : X → I (L) (for short, t ∗ ) by letting 1; s¡t; ∗ t (x)(s+) = for all x ∈ X: (1.2) 0; s¿t It is easy to verify that t60; 1; P ∗ (x)(t−) = P(x); 0¡t61; 0; t¿1;
∗
t (x)(s−) =
1; 0;
s6t; for all x ∈ X: s¿t
(1.3)
(1.4)
Theorem 1.1 (Wang [6]). If ∈ I (L) X ; then we have (t ∗ ∧ t ()∗ ): = t∈[0;1)
Theorem 1.2 (Wang [6]). If ∈ I (L) X ; P ∈ LX and t ∈ I; then the following equalities hold: (1) (t ()) = !1−t ( ); (2) (!t ()) = 1−t ( );
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
(3) (P ∗ ) = (P )∗ ; (4) t (P ∗ ) = P for all t ∈ [0; 1); (5) !t (P ∗ ) = P for all t ∈ [0; 1).
there are a P ∈ − (x ) and a t¿0 such that kP ⊂ W for all k with |k − k0 |¡t. X
Theorem 1.3 (Wang [6]). Let (L ; ) be an L-fts, ∈ I (L) X ; P ∈ LX . Then (1) ∈ !I (L) () i: t () ∈ for all t ∈ I ; (2) ∈ !I (L) () i: (!t ()) ∈ for all t ∈ I ; (3) P ∈ i: P ∗ ∈ !I (L) (). For each ∈ L follows: 1; ; t (s+) = ; 0;
295
and t ∈ I , we deJne ; t ∈ I (L) as
Lemma 2.1. Let ∈ I (L) X and t ∈ I . Then (1) (t ∗ ) = (1 − t)∗ ; (2) t1 ¿t2 ⇒ !t1 ()6!t2 (); (3) t1 ¿t2 ⇒ t1 ()6t2 (). Proof. (1) By (1.2) and (1.4), we have (t ∗ ) (x)(s+) = [t ∗ (x)] (s+) = [t ∗ (x)((1 − s)−)] = (1 − t)∗ (x)(s+)
s¡0; 06s¡t; s¿t:
for all x ∈ X; s ∈ R: (1.5)
Theorem 1.4 (Wang [6]). Let ∈ I (L). Then is a molecule in I (L) i: there exists an ∈ M (L) and t ∈ I such that = ; t .
2. Induced I (L)-fuzzy topological vector spaces In this section, the main result of the present study will be shown. We begin by recalling the deJnition of L-fuzzy topological vector space. Denition 2.1 (Fang [1]). Let X be a vector space over the Jeld K (K = R or C) and be an L-fuzzy topology on X . The pair (LX ; ) is said to be an L-fuzzy topological vector space (for short, L-ftvs) if the following two mappings are continuous: (1) f : X × X → X; (x; y) → x + y; (2) g : K × X → X; (k; x) → k x; where X × X and K × X are equipped with the corresponding product L-fuzzy topologies × and JK × , respectively. Remark 2.1. It is not diNcult to show that (cf. [1, Propositions 2.1 and 2.2]). (1) the mapping f is continuous if and only if for each x; y ∈ X; ∈ M (L) and each W ∈ − ((x + y) ), there are a P ∈ − (x ) and a Q ∈ − (y ) such that P + Q ⊂ W ; (2) the mapping g is continuous if and only if for each x ∈ M (LX ); k0 ∈ K and each W ∈ − ((k0 x) ),
Hence (t ∗ ) = (1 − t)∗ . Conditions (2) and (3) are obvious. Lemma 2.2. Let X be a vector space over K. P; Q ∈ LX ; ; ! ∈ I (L) X ; and t ∈ I . Then the following equalities hold: (1) t ∗ ∧ (P ∗ + Q∗ ) = (t ∗ ∧ P ∗ ) + (t ∗ ∧ Q∗ ); (2) t ( + !) = t () + t (!); (3) !t ( + !) = !t () + !t (!). Proof. (1) For any x ∈ X and any s ∈ I , if s¡t, then [t ∗ ∧ (P ∗ + Q∗ )](x)(s+) = t ∗ (x)(s+) ∧ (P ∗ (y)(s+) ∧ Q∗ (z)(s+)) y+z=x
=
[P ∗ (y)(s+) ∧ Q∗ (z)(s+)]
y+z=x
=
[(t ∗ (y)(s+) ∧ P ∗ (y)(s+))
y+z=x
∧ (t ∗ (z)(s+) ∧ Q∗ (z)(s+))] = [(t ∗ ∧ P ∗ ) + (t ∗ ∧ Q∗ )](x)(s+): If s¿t, then we have [t ∗ ∧ (P ∗ + Q∗ )](x)(s+) = 0 = [(t ∗ ∧ P ∗ + (t ∗ ∧ Q∗ )](x)(s+): Hence (1) holds.
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J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
(2) We Jrst prove the following equality: [(y)(s) ∧ !(z)(s)]
Lemma 2.3. Let X be a vector space over K; P; Q ∈ LX ; and k ∈ K. Then (1) (P + Q)∗ = P ∗ + Q∗ . (2) (kP)∗ = kP ∗ .
s¿t
=
(y)(s) ∧
s¿t
!(z)(s) :
(2.1)
s¿t
In fact, it is obvious that [(y)(s) ∧ !(z)(s)]
Lemma 2.4. Let ∈ I (L) X ; and s ∈ I . Then for any k ∈K (1) ks∗ 6s∗ . (2) s (k) = ks ().
s¿t
6
(y)(s) ∧
s¿t
!(z)(s) :
s¿t
(y)(s)] ∧ On
the other hand, suppose that [ s¿t [ s¿t !(z)(s)] = a, then for each # ∈ ∗ (a), we have s1 ; s2 ¿t such that (y)(s1 )¿#; !(z)(s2 )¿#. Put r = min{s1 ; s2 }. Notice that (y)(·) and !(z)(·) are decreasing. We have [(y)(s) ∧ !(z)(s)]¿(y)(r) ∧ !(z)(r)¿#: s¿t
By the arbitrariness of # ∈ ∗ (a), we get ∧ !(z)(s)]¿a. Hence (2.1) holds. From (2.1) it follows that
= ( + !)(x)(t+) =
s¿t
[(y)(s)
=
=
y+z=x
=
( + !)(x)(s)
If k = 0 and x = & (where & is the zero element in X ), then 0s ()(x) = 0 = s (0)(x); If k = 0 and x = &, (where & is the zero element in X ), then s ()(z) = (z)(s+) 0s ()(&) =
z∈X
(z) (s+) = (0)(&)(s+)
z∈X
= s (0)(&):
[(y)(s) ∧ !(z)(s)]
Condition (2) is proved. [(y)(s) ∧ !(z)(s)]
(y)(s)
∧
s¿t
s¿t
[(y)(s+) ∧ !(z)(s+)]
y+z=x
=
= s ()(x=k) = (ks ())(x):
=
s¿t
y+z=x s¿t
s (k)(x) = (k)(x)(s+) = (x=k)(s+)
z∈X
s¿t y+z=x
Proof. (1) If k = 0, then for any x ∈ X (ks∗ )(x) = ∗ ∗ s∗ (x=k)
∗= s (x); If k = 0,∗ then for any x∗∈ X ∗, (0s )(x) 6 {s (u) | u ∈ X } = s (x). Hence k s 6s . (2) If k = 0, then for any x ∈ X
t ( + !)(x)
=
Proof. It follows from DeJnition 1.5 immediately.
[t ()(y) ∧ t (!)(z)]
y+z=x
= (t () + t (!))(x): This completes the proof (2). (3) Analogous to (2).
!(z)(s)
Lemma 2.5. Let − (x ) denotes the set of all closed R-neighborhoods of a molecule x in (LX ; ); and − (x ; t ) denotes the set of all closed R-neighborhoods of a molecule x ; t in (I (L) X ; !I (L) ()). Then (1) If P ∈ − (x ); then for any t ∈ (0; 1] and s ∈ [0; t); P ∗ ∨ s∗ ∈ − (x ; t ). (2) If P ∈ − (x ; t ); then there exists a s ∈ (0; t) such that !s (P) ∈ − (x ). Proof. (1) Let P ∈ − (x ). Then for each t ∈ (0; 1] and s ∈ [0; t) we have ; t (s+) = P(x) = P ∗ (x)(s+) = (P ∗ ∨ s∗ )(x)(s+)
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
and so ; t (P ∗ ∨ s∗ )(x), i.e., x ; t P ∗ ∨ s∗ . By [6, Lemma 3.1] (P )∗ ∧ (1 − s)∗ ∈ !I (L) (), since P ∈ . Therefore P ∗ ∨ s∗ ∈ − (x ; t ). (2) Let P ∈ − (x ; t ). Then ; t P(x), and so there exists an s0 ∈ [0; t) such that = ; t (s0 +) P(x)(s0 +). Taking s ∈ (s0 ; t), we have P(x)(s−)6 P(s0 +). Hence P(x)(s−) = !s (P)(x). Since P ∈ !I (L) (), by Theorem 1.3, it is easy to see that !s (P) ∈ . Therefore !s (P) ∈ − (x ).
I (L) is continuous, where
Theorem 2.1. Let X be a vector space over K and an L-fuzzy topology on X . Then (LX ; ) is an L-ftvs i: (I (L) X ; !I (L) ()) is an I (L)-ftvs.
k(P ∗ ) = (kP )∗ 6(1−s ( ))∗
X
Proof. Necessity: Suppose that (L ; ) is an L-ftvs. We Jrst prove that the Zadeh’s type function f˜ : I (L) X ×X → I (L) X induced by the mapping f and I (L) is continuous, where f : X × X → X;
(x; y) → x + y:
(2.2)
For each x; y ∈ X , ; t ∈ M (I (L)) and ∈ − ((x + y) ; t ). By Lemma 2.5, there exists a s ∈ (0; t) such that !s () ∈ − ((x + y) ). Since (LX ; ) is an L-ftvs, by Remark 2.1 there exist P ∈ − (x ) and Q ∈ − (y ) such that P + Q 6(!s ()) . Thus, it follows from Lemma 2.3 and Theorem 1.2 that (P )∗ + (Q )∗ = (P + Q )∗ 6(!s () )∗ = (1−s ( ))∗ :
(2.3)
By Lemmas 2.1, 2.2, Theorem 1.1 and (2.3), we get (s∗ ∨ P ∗ ) + (s∗ ∨ Q∗ )
g : K × X → X;
297
(k; x) → kx:
(2.4)
For each k0 ∈ K; x ∈ X; ; t ∈ M (I (L)), and ∈ − ((k0 x) ; t ). By Lemma 2.5, there exists a s ∈ (0; t) such that !s () ∈ − ((k0 x) ). Since (LX ; ) is an L-ftvs, by Remark 2.2 there exist P ∈ − (x ) and r¿0 such that k P 6(!s ()) when |k − k0 |¡r. Using Theorem 1.2 and Lemma 2.3, we get when |k − k0 | ¡ r: (2.5) Thus from Lemmas 2.1, 2.4, Theorem 1.1 and (2.5) we have k(s∗ ∨ P ∗ ) = k[(1 − s)∗ ∧ (P ∗ ) ] 6 (1 − s)∗ ∧ k(P ∗ ) 6 (1 − s)∗ ∧ (1−s ( ))∗ 6 when |k − k0 | ¡ r: Notice that s∗ ∨ P ∗ ∈ − (x ; t ). Hence the mapping g˜ is continuous. Therefore (I (L) X ; !I (L) ()) is a I (L)-ftvs. Su
By Lemmas 2.5 and 2.1, there exists some s ∈ (0; 1) such that !s () ∈ − (x ) and !s (!) ∈ − (y ). Thus from Theorem 1.2, Lemma 2.2 and (2.6), we obtain (!s ()) + (!s (!)) = 1−s ( ) + 1−s (! )
= (1 − s)∗ ∧ (P )∗ + (1 − s)∗ ∧ (Q )∗ = (1 − s)∗ ∧ ((P )∗ + (Q )∗ )
= 1−s ( + ! )
6 (1 − s)∗ ∧ (1−s ( ))∗ 6 :
6 1−s ((W )∗ ) = W :
∗
∗
−
(2.6)
∗
∗
By Lemma 2.5, s ∨ P ∈ (x ; t ), and s ∨ Q ∈ − (y ; t ). Therefore, the mapping f˜ is continuous. Next, we prove that the Zadeh’s type function g˜ : I (L)K×X → I (L) X induced by the mapping g and
This shows that the Zadeh’s-type function f˜ induced by (2.2) and L is continuous. For each k0 ∈ K, x ∈ X , ∈ M (L) and W ∈ − ((k0 x) ), by Lemma 2.5, W ∗ ∈ − ((k0 x) ;1 ). Since (I (L) X ; !I (L) ()) is an I (L)-ftvs, there exist
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J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
∈ − (x ;1 ) and r¿0 such that k 6(W ∗ ) when |k − k0 |¡r. Since ∈ − (x ; 1 ), by Lemma 2.5 there exists s ∈ (0; 1) such that !s () ∈ − (x ). Thus, from Theorem 1.2 and Lemma 2.4, we obtain k(!s ()) = k1−s ( ) = 1−s (k ) 6 1−s ((W )∗ ) = W ; when |k − k0 | ¡ r: This implies that the Zadeh’s-type function g˜ induced by (2.4) and L is continuous. Therefore (LX ; ) is an L-ftvs. In particular, letting L = {0; 1} in Theorem 2.1, we obtain the following: Corollary 2.1. Let T be a topology on the linear space X . Then (X; T ) is a topological vector space i: induced space (I X ; !(J )) is a fuzzy topological vector space [3,8]; where !(J ) denotes the set of all lower semicontinuous mappings on (X; T ). 3. Product and quotient spaces Xi Lemma 3.1 (Xu and Fang [9]). Let {(L
; i )}i∈( be a family of L-ftvses; X = i∈( Xi ; = i∈( i . Then (LX ; ) is again an L-ftvs.
Lemma 3.2 (Xu and Fang [9]). Let (LX ; ) be an L-ftvs and E a linear subspace of X . Let X=E = {x˜ | x ∈ X } be the quotient space; where x˜ = {y ∈ X | y − x ∈ E}. Put =E = {q(G) | G ∈ }; where q : X → X=E; x → x. ˜ Then (LX=E ; =E) is again an L-ftvs. Lemma 3.3. Let (LX ; ) be an L-ftvs and let f : X → Y be a surjective mapping. Then f(!I (L) ()) = !I (L) (f()): Proof. Assume that ∈ f(!I (L) ()). Then there exists an ! ∈ !I (L) () such that = f(!). By Theorem 1.3(1), for each t ∈ I , we have t (!) ∈ . Thus t () = t (f(!)) = f(t (!)) ∈ f(). Hence ∈ !I (L) (f()).
Conversely, assume that ∈ !I (L) (f()). Then for any t ∈ I , we have t () ∈ f(), and so there exists a !t ∈ such that t () = f(!t ). Let ! = t∈I (t ∗ ∧ !∗t ). Then ! ∈ !I (L) () by Wang [6, Lemma 3.1]. Noting that f is surjective, it is not diNcult to verify that f(t ∗ ) = tY∗ (t ∈ I ) and f(P ∗ ) = f(P)∗ (P ∈ LX ). Thus, we have f(!) =
[f(t ∗ ) ∧ f(!∗t )] =
t∈I
[tY∗ ∧ t ()∗ ] = :
t∈I
Hence ∈ f(!I (L) ()). Xi Theorem 3.1. Let {(I (L) ; -i )}i∈( be a family of induced I (L)-ftvses; X = i∈( Xi ; - = i∈( -i . Then (I (L) X ; -) is an induced I (L)-ftvs.
Proof. Since (I (L) Xi ; -i ) is an induced I (L)-ftvs, there exists an L-fuzzy vector topology i such that -i = !I (L) (i ) (i ∈ (). By Wang [6, Theorem 3.1], we have -=
!I (L) (i ) = !I (L)
i∈(
i
:
i∈(
Since (LX ; i ) (i ∈ () is an L-ftvs, by Lemma 3.1 (LX ; i∈( i ) is an L-ftvs. Thus, by Theorem 2.1 (I (L) X ; -) is an induced I (L)-ftvs. Theorem 3.2. Let (I (L) X ; -) be an induced I (L)-ftvs and E a linear subspace of X . Then the quotient space (I (L) X=E ; -=E) is again an induced I (L)-ftvs. Proof. Since (I (L) X ; -) is an induced I (L)-ftvs, there exists an L-fuzzy vector topology on X such that - = !I (L) (). So (LX ; ) is an L-ftvs by Theorem 2.1. By Lemma 3.2, (LX=E ; =E) is also an L-ftvs. Again by Theorem 2.1, (I (L) X=E ; !I (L) (=E)) is an I (L)-ftvs. Besides, by Lemma 3.3 we have -=E = q(-) = q(!I (L) ()) = !I (L) (q()) = !I (L) (=E): Thus, the quotient space (I (L) X=E ; -=E) is an induced I (L)-ftvs.
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
Acknowledgements The authors are grateful to the referees for their valuable comments and suggestions. References [1] Fang Jin-Xuan, Yan Cong-Hua, L-fuzzy topological vector spaces, J. Fuzzy Math. 5 (1) (1997) 133–144. [2] Fang Jin-Xuan, The continuity of fuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997) 829–838. [3] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets and Systems 6 (1981) 85–95. [4] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141–153.
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[5] Liu Ying-Ming, Structures of fuzzy order-homomorphisms, Fuzzy Sets and Systems 21 (1987) 43–51. [6] Wang Ge-Ping, Induced I (L)-fuzzy topological spaces, Fuzzy Sets and Systems 43 (1991) 69–80. [7] Wang Guo-Jun, Theory of L-Fuzzy Topological Spaces, Shanxi Normal University Publishing House, Shanxi, 1988 (in Chinese). [8] Wu Cong-Xin, Fang Jin-Xuan, RedeJne of fuzzy topological vector spaces, Sci. Exploration 2 (4) (1982) 113–116 (in Chinese). [9] Xu Xiao-Li, Fang Jin-Xuan, Product spaces and quotient spaces of L-fuzzy topological vector spaces, Fuzzy Sets Theory and Applications, Hebi University Publishing House, Baoding, 1998, pp. 20 –22 (in Chinese). [10] Yan Cong-Hua, Two generating mappings !L and .L in complete lattices TVS and LFTVS, J. Fuzzy Math. 6 (3) (1998) 745–772.
www.elsevier.com/locate/fss
Induced I (L)-fuzzy topological vector spaces Jin-Xuan Fang ∗ , Cong-Hua Yan Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu, 210097, People’s Republic of China Received 28 December 1998; received in revised form 7 January 2000; accepted 14 February 2000
Abstract In this paper we introduce and study the induced I (L)-fuzzy topological vector spaces. The following main result is proved: If (LX ; ) is an L-fuzzy topological vector space, then (I (L)X ; !I (L) ()) is an I (L)-fuzzy topological vector space, where I (L) is the fuzzy unit interval and !I (L) () is the induced I (L)-fuzzy topology of the L-fuzzy topology . Furthermore, we also prove that the induced I (L)-fuzzy topological vector spaces (I (L)X ; !I (L) ()) preserve the product and quotient space c 2001 Elsevier Science B.V. All rights reserved. as well. Keywords: Fuzzy unit interval; Induced I (L)-fuzzy topology; L-fuzzy topological vector spaces; Induced I (L)-fuzzy topological vector spaces
1. Introduction and preliminaries In [6], Wang introduced the concept of induced I (L)-fuzzy topological spaces by using the I (L)valued lower semicontinuous mappings [4]. Let (LX ; ) be an L-fuzzy topological space. !I (L) () denotes the set of all I (L)-valued lower semicontinuous mappings on X . Then (I (L) X ; !I (L) ()) is an I (L)-fuzzy topological space and is called the induced I (L)-fuzzy topological space. Recently, we introduced and studied L-fuzzy topological vector spaces (see [1,2,10]). In [10], we proved that (X; T ) is a topological vector space i= (LX ; !L (T )) is an L-fuzzy topological vector space, where !L (T ) is the Project supported by the National Natural Science Foundation of China. ∗ Corresponding author. E-mail address: [email protected] (J.-X. Fang).
set of all L-valued lower semicontinuous mappings on X . Thus, a natural question can be presented: If (LX ; ) is an L-fuzzy topological vector space, is (I (L) X ; !I (L) ()) an I (L)-fuzzy topological vector space? In this paper we shall give a positive answer to this question. We prove that if is an L-fuzzy topology on the vector space X , then (I (L) X ; !I (L) ()) is an I (L)-fuzzy topological vector space i= (LX ; ) is an L-fuzzy topological vector space. Furthermore, we also show that the induced I (L)-fuzzy topological vector space (I (L) X ; !I (L) ()) preserves the product and the quotient space. Throughout this paper, L will denote a fuzzy lattice, i.e. a completely distributive lattice [5] with orderreversing involution → . 0 and 1 are its smallest and greatest elements, respectively. M (L) denotes the set of nonzero union irreducible elements in L. The elements of M (L) are also called molecules [7] in L.
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 3 7 - 3
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For a ∈ L; ∗ (a) denotes a standard minimal family [7] of a. Given a nonempty set X; LX denotes the collection of all L-fuzzy sets on X . X denotes an L-fuzzy set which takes the constant value on X . Let I denote the unit interval [0; 1]. The L-fuzzy unit interval I (L) is the set of all equivalence classes [ ], where : R → L is a monotone decreasing mapping satisfying (t) = 1 for t¡0 and (t) = 0 for t¿1, and ∈ [ ] i= (t−) = (t−) and (t+) = (t+), for all t ∈ I . The natural L-fuzzy topology on I (L) is generated from the subbase {Lt ; Rt | t ∈ I }, where Lt [ ] = (t−) , and Rt [ ] = (t+). A partial order on I (L) is naturally deJned by [ ]6[] i= (t−)6(t−) and (t+)6(t+), for all t ∈ I . Let [ ] ∈ I (L), we deJne [ ] = [ ], where ∈ LR is deJned by (t) = (1 − t) for all t ∈ R. To simplify notation, we shall identify equivalence classes with their representatives in the following. Let ∈ I (L), and t ∈ R, then we have (t−) = (t+) =
{ (s+) | s ¡ t};
Denition 1.3 (Kubiak [4] and Wang [6]). For each t ∈ I , deJne mappings t ; !t : I (L) X → LX as follows: t () = −1 (Rt ); for all ∈ I (L)X : Obviously, t ()(x) = (x)(t+);
Denition 1.1 (Fang and Yan [1]). An L-fuzzy topology on X is a subfamily with the following properties: (1) X ∈ for every ∈ L; (2) is closed under arbitrary unions; (3) is closed under Jnite intersections. The pair (LX ; ) is called an L-fuzzy topological space (for short, L-fts). We write = {A | A ∈ }. The member of is called open L-fuzzy set and the member of is called closed L-fuzzy set. Denition 1.2 (Wang [7]). Let (LX ; ) be an L-fts, x ∈ M ∗ (LX ). A ∈ LX is called an R-neighborhood of x if there exists a P ∈ such that x ∈= P and A ⊂ P. Obviously, if P ∈ and x ∈= P, then P is a closed R-neighborhood of x . The set of all closed R-neighborhoods of x is denoted by − (x ).
!t ()(x) = (x)(t−):
Denition 1.4 (Kubiak [4] and Wang [6]). Let (LX ; ) be an L-fts. A mapping : X → I (L) is called I (L)-valued lower (resp. upper) semicontinuous if for each t ∈ I; t () ∈ (resp. (!t ()) ∈ ). Denition 1.5 (Wang [6]). DeJne the mapping ∗ : LX → I (L) X satisfying P ∗ (x)(t+) 1; = P(x); 0;
{ (s−) | s ¿ t}:
From this it is easy to prove that the following statements are equivalent: (1) ; ∈ I (L); = ; (2) (t+) = (t+) for all t ∈ I ; (3) (t−) = (t−) for all t ∈ I .
!t () = −1 (Lt )
t¡0; 06t¡1; t¿1
(1.1)
for all P ∈ LX ; x ∈ X: Moreover, for each t ∈ R, deJne a constant mapping tX∗ : X → I (L) (for short, t ∗ ) by letting 1; s¡t; ∗ t (x)(s+) = for all x ∈ X: (1.2) 0; s¿t It is easy to verify that t60; 1; P ∗ (x)(t−) = P(x); 0¡t61; 0; t¿1;
∗
t (x)(s−) =
1; 0;
s6t; for all x ∈ X: s¿t
(1.3)
(1.4)
Theorem 1.1 (Wang [6]). If ∈ I (L) X ; then we have (t ∗ ∧ t ()∗ ): = t∈[0;1)
Theorem 1.2 (Wang [6]). If ∈ I (L) X ; P ∈ LX and t ∈ I; then the following equalities hold: (1) (t ()) = !1−t ( ); (2) (!t ()) = 1−t ( );
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
(3) (P ∗ ) = (P )∗ ; (4) t (P ∗ ) = P for all t ∈ [0; 1); (5) !t (P ∗ ) = P for all t ∈ [0; 1).
there are a P ∈ − (x ) and a t¿0 such that kP ⊂ W for all k with |k − k0 |¡t. X
Theorem 1.3 (Wang [6]). Let (L ; ) be an L-fts, ∈ I (L) X ; P ∈ LX . Then (1) ∈ !I (L) () i: t () ∈ for all t ∈ I ; (2) ∈ !I (L) () i: (!t ()) ∈ for all t ∈ I ; (3) P ∈ i: P ∗ ∈ !I (L) (). For each ∈ L follows: 1; ; t (s+) = ; 0;
295
and t ∈ I , we deJne ; t ∈ I (L) as
Lemma 2.1. Let ∈ I (L) X and t ∈ I . Then (1) (t ∗ ) = (1 − t)∗ ; (2) t1 ¿t2 ⇒ !t1 ()6!t2 (); (3) t1 ¿t2 ⇒ t1 ()6t2 (). Proof. (1) By (1.2) and (1.4), we have (t ∗ ) (x)(s+) = [t ∗ (x)] (s+) = [t ∗ (x)((1 − s)−)] = (1 − t)∗ (x)(s+)
s¡0; 06s¡t; s¿t:
for all x ∈ X; s ∈ R: (1.5)
Theorem 1.4 (Wang [6]). Let ∈ I (L). Then is a molecule in I (L) i: there exists an ∈ M (L) and t ∈ I such that = ; t .
2. Induced I (L)-fuzzy topological vector spaces In this section, the main result of the present study will be shown. We begin by recalling the deJnition of L-fuzzy topological vector space. Denition 2.1 (Fang [1]). Let X be a vector space over the Jeld K (K = R or C) and be an L-fuzzy topology on X . The pair (LX ; ) is said to be an L-fuzzy topological vector space (for short, L-ftvs) if the following two mappings are continuous: (1) f : X × X → X; (x; y) → x + y; (2) g : K × X → X; (k; x) → k x; where X × X and K × X are equipped with the corresponding product L-fuzzy topologies × and JK × , respectively. Remark 2.1. It is not diNcult to show that (cf. [1, Propositions 2.1 and 2.2]). (1) the mapping f is continuous if and only if for each x; y ∈ X; ∈ M (L) and each W ∈ − ((x + y) ), there are a P ∈ − (x ) and a Q ∈ − (y ) such that P + Q ⊂ W ; (2) the mapping g is continuous if and only if for each x ∈ M (LX ); k0 ∈ K and each W ∈ − ((k0 x) ),
Hence (t ∗ ) = (1 − t)∗ . Conditions (2) and (3) are obvious. Lemma 2.2. Let X be a vector space over K. P; Q ∈ LX ; ; ! ∈ I (L) X ; and t ∈ I . Then the following equalities hold: (1) t ∗ ∧ (P ∗ + Q∗ ) = (t ∗ ∧ P ∗ ) + (t ∗ ∧ Q∗ ); (2) t ( + !) = t () + t (!); (3) !t ( + !) = !t () + !t (!). Proof. (1) For any x ∈ X and any s ∈ I , if s¡t, then [t ∗ ∧ (P ∗ + Q∗ )](x)(s+) = t ∗ (x)(s+) ∧ (P ∗ (y)(s+) ∧ Q∗ (z)(s+)) y+z=x
=
[P ∗ (y)(s+) ∧ Q∗ (z)(s+)]
y+z=x
=
[(t ∗ (y)(s+) ∧ P ∗ (y)(s+))
y+z=x
∧ (t ∗ (z)(s+) ∧ Q∗ (z)(s+))] = [(t ∗ ∧ P ∗ ) + (t ∗ ∧ Q∗ )](x)(s+): If s¿t, then we have [t ∗ ∧ (P ∗ + Q∗ )](x)(s+) = 0 = [(t ∗ ∧ P ∗ + (t ∗ ∧ Q∗ )](x)(s+): Hence (1) holds.
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(2) We Jrst prove the following equality: [(y)(s) ∧ !(z)(s)]
Lemma 2.3. Let X be a vector space over K; P; Q ∈ LX ; and k ∈ K. Then (1) (P + Q)∗ = P ∗ + Q∗ . (2) (kP)∗ = kP ∗ .
s¿t
=
(y)(s) ∧
s¿t
!(z)(s) :
(2.1)
s¿t
In fact, it is obvious that [(y)(s) ∧ !(z)(s)]
Lemma 2.4. Let ∈ I (L) X ; and s ∈ I . Then for any k ∈K (1) ks∗ 6s∗ . (2) s (k) = ks ().
s¿t
6
(y)(s) ∧
s¿t
!(z)(s) :
s¿t
(y)(s)] ∧ On
the other hand, suppose that [ s¿t [ s¿t !(z)(s)] = a, then for each # ∈ ∗ (a), we have s1 ; s2 ¿t such that (y)(s1 )¿#; !(z)(s2 )¿#. Put r = min{s1 ; s2 }. Notice that (y)(·) and !(z)(·) are decreasing. We have [(y)(s) ∧ !(z)(s)]¿(y)(r) ∧ !(z)(r)¿#: s¿t
By the arbitrariness of # ∈ ∗ (a), we get ∧ !(z)(s)]¿a. Hence (2.1) holds. From (2.1) it follows that
= ( + !)(x)(t+) =
s¿t
[(y)(s)
=
=
y+z=x
=
( + !)(x)(s)
If k = 0 and x = & (where & is the zero element in X ), then 0s ()(x) = 0 = s (0)(x); If k = 0 and x = &, (where & is the zero element in X ), then s ()(z) = (z)(s+) 0s ()(&) =
z∈X
(z) (s+) = (0)(&)(s+)
z∈X
= s (0)(&):
[(y)(s) ∧ !(z)(s)]
Condition (2) is proved. [(y)(s) ∧ !(z)(s)]
(y)(s)
∧
s¿t
s¿t
[(y)(s+) ∧ !(z)(s+)]
y+z=x
=
= s ()(x=k) = (ks ())(x):
=
s¿t
y+z=x s¿t
s (k)(x) = (k)(x)(s+) = (x=k)(s+)
z∈X
s¿t y+z=x
Proof. (1) If k = 0, then for any x ∈ X (ks∗ )(x) = ∗ ∗ s∗ (x=k)
∗= s (x); If k = 0,∗ then for any x∗∈ X ∗, (0s )(x) 6 {s (u) | u ∈ X } = s (x). Hence k s 6s . (2) If k = 0, then for any x ∈ X
t ( + !)(x)
=
Proof. It follows from DeJnition 1.5 immediately.
[t ()(y) ∧ t (!)(z)]
y+z=x
= (t () + t (!))(x): This completes the proof (2). (3) Analogous to (2).
!(z)(s)
Lemma 2.5. Let − (x ) denotes the set of all closed R-neighborhoods of a molecule x in (LX ; ); and − (x ; t ) denotes the set of all closed R-neighborhoods of a molecule x ; t in (I (L) X ; !I (L) ()). Then (1) If P ∈ − (x ); then for any t ∈ (0; 1] and s ∈ [0; t); P ∗ ∨ s∗ ∈ − (x ; t ). (2) If P ∈ − (x ; t ); then there exists a s ∈ (0; t) such that !s (P) ∈ − (x ). Proof. (1) Let P ∈ − (x ). Then for each t ∈ (0; 1] and s ∈ [0; t) we have ; t (s+) = P(x) = P ∗ (x)(s+) = (P ∗ ∨ s∗ )(x)(s+)
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
and so ; t (P ∗ ∨ s∗ )(x), i.e., x ; t P ∗ ∨ s∗ . By [6, Lemma 3.1] (P )∗ ∧ (1 − s)∗ ∈ !I (L) (), since P ∈ . Therefore P ∗ ∨ s∗ ∈ − (x ; t ). (2) Let P ∈ − (x ; t ). Then ; t P(x), and so there exists an s0 ∈ [0; t) such that = ; t (s0 +) P(x)(s0 +). Taking s ∈ (s0 ; t), we have P(x)(s−)6 P(s0 +). Hence P(x)(s−) = !s (P)(x). Since P ∈ !I (L) (), by Theorem 1.3, it is easy to see that !s (P) ∈ . Therefore !s (P) ∈ − (x ).
I (L) is continuous, where
Theorem 2.1. Let X be a vector space over K and an L-fuzzy topology on X . Then (LX ; ) is an L-ftvs i: (I (L) X ; !I (L) ()) is an I (L)-ftvs.
k(P ∗ ) = (kP )∗ 6(1−s ( ))∗
X
Proof. Necessity: Suppose that (L ; ) is an L-ftvs. We Jrst prove that the Zadeh’s type function f˜ : I (L) X ×X → I (L) X induced by the mapping f and I (L) is continuous, where f : X × X → X;
(x; y) → x + y:
(2.2)
For each x; y ∈ X , ; t ∈ M (I (L)) and ∈ − ((x + y) ; t ). By Lemma 2.5, there exists a s ∈ (0; t) such that !s () ∈ − ((x + y) ). Since (LX ; ) is an L-ftvs, by Remark 2.1 there exist P ∈ − (x ) and Q ∈ − (y ) such that P + Q 6(!s ()) . Thus, it follows from Lemma 2.3 and Theorem 1.2 that (P )∗ + (Q )∗ = (P + Q )∗ 6(!s () )∗ = (1−s ( ))∗ :
(2.3)
By Lemmas 2.1, 2.2, Theorem 1.1 and (2.3), we get (s∗ ∨ P ∗ ) + (s∗ ∨ Q∗ )
g : K × X → X;
297
(k; x) → kx:
(2.4)
For each k0 ∈ K; x ∈ X; ; t ∈ M (I (L)), and ∈ − ((k0 x) ; t ). By Lemma 2.5, there exists a s ∈ (0; t) such that !s () ∈ − ((k0 x) ). Since (LX ; ) is an L-ftvs, by Remark 2.2 there exist P ∈ − (x ) and r¿0 such that k P 6(!s ()) when |k − k0 |¡r. Using Theorem 1.2 and Lemma 2.3, we get when |k − k0 | ¡ r: (2.5) Thus from Lemmas 2.1, 2.4, Theorem 1.1 and (2.5) we have k(s∗ ∨ P ∗ ) = k[(1 − s)∗ ∧ (P ∗ ) ] 6 (1 − s)∗ ∧ k(P ∗ ) 6 (1 − s)∗ ∧ (1−s ( ))∗ 6 when |k − k0 | ¡ r: Notice that s∗ ∨ P ∗ ∈ − (x ; t ). Hence the mapping g˜ is continuous. Therefore (I (L) X ; !I (L) ()) is a I (L)-ftvs. Su
By Lemmas 2.5 and 2.1, there exists some s ∈ (0; 1) such that !s () ∈ − (x ) and !s (!) ∈ − (y ). Thus from Theorem 1.2, Lemma 2.2 and (2.6), we obtain (!s ()) + (!s (!)) = 1−s ( ) + 1−s (! )
= (1 − s)∗ ∧ (P )∗ + (1 − s)∗ ∧ (Q )∗ = (1 − s)∗ ∧ ((P )∗ + (Q )∗ )
= 1−s ( + ! )
6 (1 − s)∗ ∧ (1−s ( ))∗ 6 :
6 1−s ((W )∗ ) = W :
∗
∗
−
(2.6)
∗
∗
By Lemma 2.5, s ∨ P ∈ (x ; t ), and s ∨ Q ∈ − (y ; t ). Therefore, the mapping f˜ is continuous. Next, we prove that the Zadeh’s type function g˜ : I (L)K×X → I (L) X induced by the mapping g and
This shows that the Zadeh’s-type function f˜ induced by (2.2) and L is continuous. For each k0 ∈ K, x ∈ X , ∈ M (L) and W ∈ − ((k0 x) ), by Lemma 2.5, W ∗ ∈ − ((k0 x) ;1 ). Since (I (L) X ; !I (L) ()) is an I (L)-ftvs, there exist
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∈ − (x ;1 ) and r¿0 such that k 6(W ∗ ) when |k − k0 |¡r. Since ∈ − (x ; 1 ), by Lemma 2.5 there exists s ∈ (0; 1) such that !s () ∈ − (x ). Thus, from Theorem 1.2 and Lemma 2.4, we obtain k(!s ()) = k1−s ( ) = 1−s (k ) 6 1−s ((W )∗ ) = W ; when |k − k0 | ¡ r: This implies that the Zadeh’s-type function g˜ induced by (2.4) and L is continuous. Therefore (LX ; ) is an L-ftvs. In particular, letting L = {0; 1} in Theorem 2.1, we obtain the following: Corollary 2.1. Let T be a topology on the linear space X . Then (X; T ) is a topological vector space i: induced space (I X ; !(J )) is a fuzzy topological vector space [3,8]; where !(J ) denotes the set of all lower semicontinuous mappings on (X; T ). 3. Product and quotient spaces Xi Lemma 3.1 (Xu and Fang [9]). Let {(L
; i )}i∈( be a family of L-ftvses; X = i∈( Xi ; = i∈( i . Then (LX ; ) is again an L-ftvs.
Lemma 3.2 (Xu and Fang [9]). Let (LX ; ) be an L-ftvs and E a linear subspace of X . Let X=E = {x˜ | x ∈ X } be the quotient space; where x˜ = {y ∈ X | y − x ∈ E}. Put =E = {q(G) | G ∈ }; where q : X → X=E; x → x. ˜ Then (LX=E ; =E) is again an L-ftvs. Lemma 3.3. Let (LX ; ) be an L-ftvs and let f : X → Y be a surjective mapping. Then f(!I (L) ()) = !I (L) (f()): Proof. Assume that ∈ f(!I (L) ()). Then there exists an ! ∈ !I (L) () such that = f(!). By Theorem 1.3(1), for each t ∈ I , we have t (!) ∈ . Thus t () = t (f(!)) = f(t (!)) ∈ f(). Hence ∈ !I (L) (f()).
Conversely, assume that ∈ !I (L) (f()). Then for any t ∈ I , we have t () ∈ f(), and so there exists a !t ∈ such that t () = f(!t ). Let ! = t∈I (t ∗ ∧ !∗t ). Then ! ∈ !I (L) () by Wang [6, Lemma 3.1]. Noting that f is surjective, it is not diNcult to verify that f(t ∗ ) = tY∗ (t ∈ I ) and f(P ∗ ) = f(P)∗ (P ∈ LX ). Thus, we have f(!) =
[f(t ∗ ) ∧ f(!∗t )] =
t∈I
[tY∗ ∧ t ()∗ ] = :
t∈I
Hence ∈ f(!I (L) ()). Xi Theorem 3.1. Let {(I (L) ; -i )}i∈( be a family of induced I (L)-ftvses; X = i∈( Xi ; - = i∈( -i . Then (I (L) X ; -) is an induced I (L)-ftvs.
Proof. Since (I (L) Xi ; -i ) is an induced I (L)-ftvs, there exists an L-fuzzy vector topology i such that -i = !I (L) (i ) (i ∈ (). By Wang [6, Theorem 3.1], we have -=
!I (L) (i ) = !I (L)
i∈(
i
:
i∈(
Since (LX ; i ) (i ∈ () is an L-ftvs, by Lemma 3.1 (LX ; i∈( i ) is an L-ftvs. Thus, by Theorem 2.1 (I (L) X ; -) is an induced I (L)-ftvs. Theorem 3.2. Let (I (L) X ; -) be an induced I (L)-ftvs and E a linear subspace of X . Then the quotient space (I (L) X=E ; -=E) is again an induced I (L)-ftvs. Proof. Since (I (L) X ; -) is an induced I (L)-ftvs, there exists an L-fuzzy vector topology on X such that - = !I (L) (). So (LX ; ) is an L-ftvs by Theorem 2.1. By Lemma 3.2, (LX=E ; =E) is also an L-ftvs. Again by Theorem 2.1, (I (L) X=E ; !I (L) (=E)) is an I (L)-ftvs. Besides, by Lemma 3.3 we have -=E = q(-) = q(!I (L) ()) = !I (L) (q()) = !I (L) (=E): Thus, the quotient space (I (L) X=E ; -=E) is an induced I (L)-ftvs.
J.-X. Fang, C.-H. Yan / Fuzzy Sets and Systems 121 (2001) 293–299
Acknowledgements The authors are grateful to the referees for their valuable comments and suggestions. References [1] Fang Jin-Xuan, Yan Cong-Hua, L-fuzzy topological vector spaces, J. Fuzzy Math. 5 (1) (1997) 133–144. [2] Fang Jin-Xuan, The continuity of fuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997) 829–838. [3] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets and Systems 6 (1981) 85–95. [4] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141–153.
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[5] Liu Ying-Ming, Structures of fuzzy order-homomorphisms, Fuzzy Sets and Systems 21 (1987) 43–51. [6] Wang Ge-Ping, Induced I (L)-fuzzy topological spaces, Fuzzy Sets and Systems 43 (1991) 69–80. [7] Wang Guo-Jun, Theory of L-Fuzzy Topological Spaces, Shanxi Normal University Publishing House, Shanxi, 1988 (in Chinese). [8] Wu Cong-Xin, Fang Jin-Xuan, RedeJne of fuzzy topological vector spaces, Sci. Exploration 2 (4) (1982) 113–116 (in Chinese). [9] Xu Xiao-Li, Fang Jin-Xuan, Product spaces and quotient spaces of L-fuzzy topological vector spaces, Fuzzy Sets Theory and Applications, Hebi University Publishing House, Baoding, 1998, pp. 20 –22 (in Chinese). [10] Yan Cong-Hua, Two generating mappings !L and .L in complete lattices TVS and LFTVS, J. Fuzzy Math. 6 (3) (1998) 745–772.