semi-norms, a characterization of the new locally convex L-topological vector spaces
is presented. Finally, as applications of this characterization, the Hausdorff separation
property, convergence of molecule nets and boundedness of L-fuzzy sets in locally convex
L-topological vector spaces are studied.
Keywords: Analysis; L-topological vector spaces; Locally convex L-topological vector
spaces; Generalized L-fuzzy semi-norms
1. Introduction
Fang and Yan [2] introduced the notion of L-fuzzy topological vector space in 1997.
According to the standardized terminology in [3], more accurately, it should be called
L-topological vector space (briefly, L-tvs), which is the generalization of both the notion
of classical topological vector space and that of I-topological vector space introduced by
Katsaras [4], where I denotes the real unit interval [0, 1].
It is well-known that the theory of locally convex topological vector spaces plays a
quite important role in classical functional analysis and its applications. So, it is natural
This work is supported by the National Natural Science Foundation of China (No. 10671094) and the
Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20060319001).

and necessary to find the counterpart of locally convex topological vector spaces in the
research of L-tvs. In 1999, Yan and Fang [10] introduced the concept of locally convex
L-tvs by means of a prebase and characterized such space by a family of L-probabilistic
semi-norms. Moreover, Yan [9] also characterized such space by a family of L-fuzzy pseudonorms.
However, this definition of Yan and Fang only relates to a class of L-topological
vector spaces with a prebase, and it can not contain the definition of locally convex Itopological
vector space given by Katsaras [5] as a special case. In order to overcome these
limitations, in this paper we intend to redefine and to study the locally convex L-tvs.
The present paper is organized as follows. In Section 2, we recall some basic concepts
and prove several lemmas to be needed in the sequel. In Section 3, we propose a new
definition of locally convex L-tvs, which is more reasonable than Yan and Fang’s definition.
Locally convex L-tvs defined by Yan and Fang and locally convex I-tvs defined by Katsaras
are both special cases of the new definition. In Section 4, we introduce the notion of
generalized L-fuzzy semi-norm and prove that every locally convex L-tvs defined by our
way can be characterized by a family of generalized L-fuzzy semi-norms. In Section 5, as
applications of the results obtained in Section 4, we characterize the Hausdorff separation
property, convergence of molecule nets and boundedness of L-fuzzy sets in locally convex
L-tvs by means of the family of generalized L-fuzzy semi-norms.
2. Basic concepts and lemmas
Throughout this paper, L denotes a complete and completely distributive lattice
equipped with an order-reversing involution 0 : L ! L. 0 and 1 are its bottom and
top elements, respectively. M(L) denotes the set of all non-zero union-irreducible elements
in L. The elements of M(L) are also called molecules [6] in L. For each  2 L,
() denotes the standard minimal family of . LX denotes the family of all L-fuzzy sets
on X. An L-fuzzy set which takes the constant value  2 L on X is denoted by . An
L-fuzzy set on X is called an L-fuzzy point if it takes the value 0 for all y 2 X except
one, say, x 2 X. If its value at x is  2 L \ {0}, we denote this L-fuzzy point by x. An
L-topology  on X is called stratified, if it contains all constant L-fuzzy sets on X. We
always assume that the L-topologies referred to in the present paper are all stratified and
the lattice L is regular (i.e., the intersection of each pair of non-zero elements in L is not
zero, which is equivalent to 1 2 M(L)). For other symbols which are not mentioned, we
refer to [2, 6].

x 2 M(LX). P 2 LX is called a closed R-neighborhood of x, if P 2 0 and x
 P. The
set of all closed R-neighborhoods of x is denoted by −(x).
A 2 LX is called an R-neighborhood of x, if there exists P 2 −(x) such that A 6 P.
The set of all R-neighborhoods of x is denoted by (x).
U  (x) is said to be an R-neighborhood base of x if for each P 2 (x), there
exists Q 2 U such that P 6 Q.
In the sequel, without special instruction, X always denotes a vector space over K
(K = R or C) and  denotes the zero element of X.
Definition 2.2 (Fang and Yan [2] ). The addition and the scalar multiplicative operators
of L-fuzzy sets on X are defined as follows: for A, B 2 LX and k 2 K,
(A + B)(x) = s+Wt=x
(A(s) ^ B(t)),
(kA)(x) = A(x/k), whenever k 6= 0,
(0A)(x) = 8>
<>
:
W y2X
A(y), x = ,
0, x 6= .
In particular, for L-fuzzy points x, yμ and k 2 K, we have
x + yμ = (x + y)^μ and kx = (kx).
Definition 2.3 (Fang and Yan [2] ). Let  be an L-topology on X. The pair (LX, )
is called an L-topological vector space (briefly, L-tvs) if the following two mappings (the
addition and the scalar multiplication on X):
(1) f : X × X ! X, (x, y) 7! x + y and
(2) g : K × X ! X, (k, x) 7! kx
are both continuous, where X×X and K×X are equipped with the corresponding product
L-topologies  ×  and JK × , respectively, and JK denotes the usual topology on K.
Definition 2.4 (Yan and Fang [10] ). A 2 LX is called convex if sA+(1−s)A 6 A for
all s 2 [0, 1]. A is called co-convex if A0 is convex.
Definition 2.5 (Yan and Fang [11] ). The convex hull conv(A) of an L-fuzzy set A on
X is the intersection of all the convex L-fuzzy sets on X containing A, i.e., the smallest
convex L-fuzzy set on X containing A.
Definition 2.6 (Fang and Yan [2] ). A 2 LX is called balanced if sA 6 A for all s 2 K
with |s| 6 1. If A0 is balanced, we call A co-balanced.
A is called semi-balanced if sA 6 A for all s 2 K with 0 < |s| 6 1. If A0 is semibalanced,
we call A co-semi-balanced.
P 2 LX is said to be co-normal if P() 6 P(x) for all x 2 X.
Definition 2.7 (Yan and Fang [10] ). Let (LX, ) be an L-tvs. A family B of L-fuzzy
sets on X is called a prebase of  if for each  2 M(L), B = {B0 _ μ | B 2 B,  66 μ}
is an R-neighborhood base of . An L-tvs (LX, ) is said to be locally convex, if there
exists a family B consisting of convex and balanced L-fuzzy sets on X such that B is a
prebase of .
In the case of L = [0, 1], locally convex L-tvs is exactly the locally convex I-tvs defined
by Wu and Li [8].
Lemma 2.1 (Fang and Yan [2] ). Let (LX, ) be an L-tvs, and B ( 2 M(L)) be a
closed R-neighborhood base of . Then B ( 2 M(L)) has the following properties:
(1) If W 2 B or W = , then for any x 2 X and 2 M(L) with x
 W, there
exists P 2 B such that W 6 x + P;
(2) If P, Q 2 B, then there exists W 2 B such that P _ Q 6 W;
(3) If W 2 B, then there exists P 2 B such that P0 + P0 6 W0;
(4) If W 2 B, then there exists P 2 B such that tP 0 6 W0 for all t 2 K with
|t| 6 1;
(5) If W 2 B, then for each x 2 X there exists t > 0 such that x
 tW.
Conversely, if for each  2 M(L) there is a family B of L-fuzzy sets on X satisfying the
above conditions (1)−(5), then there exists a unique L-topology  on X such that (LX, )
is an L-tvs, and B is a closed R-neighborhood base of .
Lemma 2.2 (Fang and Yan [2] ). Let (LX, ) be an L-tvs. If Q 2 , then x + Q 2  for
each x 2 X and kQ 2  for all k 2 K \ {0}.
Lemma 2.3 (Fang and Yan [2] ). Each L-tvs (LX, ) has an R-neighborhood base of 
consisting of co-balanced L-fuzzy sets on X for each  2 M(L).
Lemma 2.4 (Yan and Fang [11] ). If A 2 LX is balanced, then conv(A) is also balanced.
Lemma 2.5. Let (LX, ) be an L-tvs, P 2 LX and Q 2 . Then P + Q 2 .
Proof. If P = 0, then P + Q = 0 2 ; If P 6= 0, we let SP = {x 2 X | P(x) 6= 0}, then
it is easy to see that P = Wx2SP
xP(x). Hence
P + Q =  W x2SP

On the other hand, since  is stratified and x +Q 2  by Lemma 2.2, P(x) ^ (x +Q) 2 .
Therefore P + Q = W x2SP P(x) ^ (x + Q)2 . 
Lemma 2.6. Let (LX, ) be an L-tvs. If A 2 LX is a convex (resp. semi-balanced)
L-fuzzy set. Then its interior A is also convex (resp. semi-balanced).
Proof. We first prove that A is convex if A is.
If  = 0 or  = 1, then A + (1 − )A = 0A + A 6 1 + A = A.
If  2 (0, 1), then for any P 2  with P 6 A and any Q 2  with Q 6 A, we have
P +(1−)Q 6 A+(1−)A 6 A since A is convex, and P +(1−)Q 2  by Lemmas
2.2 and 2.5. Hence P + (1 − )Q 6 A, and so
A + (1 − )A =  _ P2, P6A
P + (1 − ) _ Q2, Q6A
Q
= _ P2, P6A
P + _ Q2, Q6A
(1 − )Q
= _ P,Q2, P,Q6A
(P + (1 − )Q) 6 A.
Therefore A is convex.
Now we prove that A is semi-balanced if A is. Let k 2 K with 0 < |k| 6 1 and P 2 
with P 6 A. Then we have kP 6 kA 6 A since A is semi-balanced and kP 2  by Lemma
2.2. Hence kP 6 A, and so kA = W P2, P6A
kP 6 A. Therefore A is semi-balanced.
This completes the proof. 
Lemma 2.7. Let (LX, ) be an L-tvs. If A 2 LX is a co-convex (resp. co-semi-balanced)
L-fuzzy set. Then its closure A− is also co-convex (resp. co-semi-balanced).
Proof. Since (A−)0 = (((A0)0)−)0 = (A0) and A0 is convex (resp. semi-balanced), (A−)0
is also convex (resp. semi-balanced) by Lemma 2.6, i.e., A− is co-convex (resp. co-semibalanced).

Lemma 2.8. P 2 LX is co-balanced iff P is co-semi-balanced and co-normal.
Proof. Without loss of generality, we may assume that P 6= 1. It suffices to show that P
is co-normal iff 0P0 6 P0.
In fact, P is co-normal () P() 6 P(x) for each x 2 X () P0(x) 6 P0() for

3. A new definition of locally convex L-tvs
In this section, we first propose a new definition of locally convex L-tvs, then discuss
the relationship between the new defined locally convex L-tvs and that in the sense of Yan
and Fang [10].
Definition 3.1. An L-tvs (LX, ) is said to be locally convex if for each  2 M(L),
there exists a family U consisting of co-convex L-fuzzy sets on X such that U is an
R-neighborhood base of .
Remark 3.1. From the discussion in Zhang and Fang [12], it is easy to see that, in the
case of L = [0, 1], the above defined locally convex L-tvs is exactly the locally convex I-tvs
in the sense of Katsaras [5].
Theorem 3.1. Let (LX, ) be an L-tvs. If (LX, ) is locally convex in the sense of Yan
and Fang, then it is also locally convex in the sense of Definition 3.1.
Proof. If (LX, ) is locally convex in the sense of Yan and Fang, then there exists a family
B consisting of convex and balanced L-fuzzy sets on X such that B is a prebase of , and
so for each  2 M(L), B = {B0 _ μ | B 2 B,  66 μ} is an R-neighborhood base of .
In the following, it suffices to show that each member of B is co-convex.
In fact, for each B 2 B and μ 2 L with 
 μ, we have (B0 _ μ)0 = B ^ μ0. On the
other hand, since B and μ0 are convex, B ^ μ0 is also convex, and so each member of B
is co-convex. This completes the proof. 
Remark 3.2. Example 3.1 in [12] shows that in the case of L = [0, 1], the converse of
the above theorem doesn’t hold in general. So, generally speaking, the converse of the
above theorem is incorrect.
Remark 3.3. From the above discussion we know that locally convex L-tvs in the sense
of Yan and Fang is a special case of that in the sense of Definition 3.1.
In the sequel, when we refer to the locally convex L-tvs, it will be always in the sense
of Definition 3.1. We rename the locally convex L-tvs in the sense of Yan and Fang as the
locally convex L-tvs with a prebase.
Theorem 3.2. Let (LX, ) be an L-tvs. Then (LX, ) is locally convex iff there exists
an R-neighborhood base of  consisting of closed, co-convex and co-balanced L-fuzzy sets
on X for each  2 M(L).

Proof. Sufficiency: It follows directly from Definition 3.1.
Necessity: For each  2 M(L), by Lemma 2.3, let B be an R-neighborhood base of
 consisting of co-balanced L-fuzzy sets. Define D = {(conv(B0))0 | B 2 B}. Then each
member of D is co-convex and co-balanced by Lemma 2.4. Moreover, (conv(B0))0 6 B
for each B 2 B, and so each member of D is an R-neighborhood of .
Next, we shall prove that D is also an R-neighborhood base of .
In fact, let Q be an arbitrary R-neighborhood of , since (LX, ) is locally convex,
then there exists a co-convex R-neighborhood P of  such that Q 6 P. On the other
hand, since B is an R-neighborhood base of , there exists B 2 B such that P 6 B,
and so Q 6 P = (conv(P0))0 6 (conv(B0))0, which implies that D is an R-neighborhood
base of .
Put V = {P− | P 2 D}. Then V is also an R-neighborhood base of  and each
member of V is closed, co-convex and co-semi-balanced by Lemma 2.7. Take U =
{Q _ Q() | Q 2 V}. To complete the proof, it suffices to show that U is an Rneighborhood
base of  and each member of U is closed, co-convex and co-balanced.
In fact, it is not difficult to see that U is an R-neighborhood base of  and each
member of U is closed, co-convex and co-normal. Moreover, for each Q 2 V, (Q _
Q())0 = Q0^(Q())0. Note that (Q())0 and Q0 are both semi-balanced, hence Q0^(Q())0
is also semi-balanced, and so Q _ Q() is co-balanced by Lemma 2.8. 
The following conclusion follows directly from Theorem 3.2.
Corollary 3.1. Let (LX, ) be a locally convex L-tvs, and U denote the set of all closed,
co-convex and co-balanced R-neighborhoods of  for each  2 M(L). Then U is a closed
R-neighborhood base of  for each  2 M(L).
4. Characterization of locally convex L-tvs
In this section, we first introduce the concept of generalized L-fuzzy semi-norm and
investigate some of its properties, then characterize locally convex L-tvs by a family of
generalized L-fuzzy semi-norms satisfying certain condition.
Definition 4.1. A mapping k · k : M(LX) ! [0,+1] is called a generalized L-fuzzy
semi-norm on X if it satisfies the following conditions:
(GSN-1) there exists  2 M(L) such that kk = 0 and kxk < +1 for all x 2 X;

(GSN-3) kx + yk 6 kxk + kyk for all x, y 2 X and  2 M(L);
(GSN-4) kxk = inf
μ2 () kxμk for each x 2 M(LX).
Remark 4.1. Since  6 μ implies ()  (μ), by (GSN-4), we have  6 μ implies
kxk > kxμk.
Remark 4.2. (1) In the case of L = [0, 1], a generalized L-fuzzy semi-norm is exactly a
generalized fuzzy semi-norm introduced by Zhang and Fang [12].
(2) L-fuzzy norm introduced by Yan and Fang [11] is a special case of generalized
L-fuzzy semi-norm.
Lemma 4.1. Let k · k be a generalized L-fuzzy semi-norm and t > 0. Define an L-fuzzy
set Uk·k,t on X by
Uk·k,t(x) =_{ 2 M(L) | kx k > t} . (4.1)
Then Uk·k,t has the following properties:
(P-1) For each x 2 M(LX), x 6 Uk·k,t () kxk > t;
(P-2) Uk·k,t = 0<Vs<t
Uk·k,s for all t > 0;
(P-3) sUk·k,t = Uk·k,st for each s, t > 0;
(P-4) (Uk·k,t)0 + (Uk·k,s)0 6 (Uk·k,t+s)0 for each s, t > 0;
(P-5) Uk·k,t() 6 Uk·k,t(x) for all t > 0, x 2 X, i.e., Uk·k,t is co-normal;
(P-6) Uk·k,t is co-balanced for each t > 0;
(P-7) For each t > 0, Uk·k,t is co-convex.
Proof. (P-1) The sufficiency is obvious. Conversely, if x 6 Uk·k,t, i.e.,  6 Uk·k,t(x),
then for each 2 () there exists μ 2 M(L) such that 6 μ and kxμk > t, and so
kx k > t by Remark 4.1. Hence kxk = inf
2 () kx k > t.
(P-2) By the definition of Uk·k,t, it is easy to see that Uk·k,t 6 Uk·k,s for each s 2 (0, t).
Hence Uk·k,t 6 0<Vs<t
Uk·k,s. On the other hand, for each x 2 M(LX) with x
 Uk·k,t,
we have kxk < t by (P-1). Hence there exists s 2 (0, t) such that kxk < s, and so
x
 Uk·k,s by (P-1). Therefore x
 0<Vs<t
Uk·k,s. This shows that 0<Vs<t
Uk·k,s 6 Uk·k,t. So
(P-2) holds.
(P-3) For each s, t > 0 and x 2 X, we have
Uk·k,st(x) =_{ 2 M(L) | kx k > st} =_{ 2 M(L) | k((1/s)x) k > t}
= Uk·k,t(x/s) = 􀀀sUk·k,t
(x).

Hence (P-3) holds.
(P-4) For each x 2 M(LX) with x
 [(Uk·k,t)0 + (Uk·k,s)0]0, i.e., 
 [(Uk·k,t)0 +
(Uk·k,s)0]0(x) = y+Vz=x
(Uk·k,t(y) _ Uk·k,s(z)), there exist a, b 2 X such that a + b = x and

 Uk·k,t(a) _ Uk·k,s(b), which implies that a
 Uk·k,t and b
 Uk·k,s, and so kak < t
and kbk < s by (P-1). Therefore kxk = ka + bk 6 kak + kbk < t + s, and so
x
 Uk·k,t+s. This completes the proof of (P-4).
(P-5) By (GSN-3), for each x 2 M(LX), we have
k k =

Udi,t _ μ whenever n 2  with n > ni. Since  is a directed set, there exists n0 2  such
that n0 > ni for each i 2 {1, 2, . . . , n}. So, if n 2  and n > n0, then (xn−x)n
 Udi,t_μ
for each i 2 {1, 2, . . . , n}, which implies (xn)n−x = (xn−x)n
 W, i.e., (xn)n
 x+W.
This completes the proof. 
Theorem 5.3. Let (LX, {k · kd}d2D) be a locally convex L-tvs, where D = S 2M(L)
D.
Then B 2 LX is -bounded iff for each d 2 D, there exist t > 0 and  2 L with 
 
such that kx kd < t for each x 2 M(LX) satisfying x
 B0 _ .
Proof. Necessity: Suppose that B 2 LX is -bounded. Note that for each d 2 D, Ud,1
is an R-neighborhood of , where Ud,1 is defined by (4.1). By Remark 5.1, there exist
t > 0 and  2 L with 
  such that Ud,t = tUd,1 6 B0 _ , and so for each x 2 M(LX)
satisfying x
 B0 _ , we have x
 Ud,t, which implies kx kd < t by (P-1).
Sufficiency: By Theorem 4.1,
U = (  n _i=1
Udi,t!_ μ
t > 0, 
 μ, di 2 D, i = 1, 2, . . . , n; n 2 N)
is an R-neighborhood base of . Hence for each W 2 (), there exist t > 0, μ 2 L with

 μ and di 2 D (i = 1, 2, . . . , n) such that W 6  nWi=1
Udi,t_ μ. It suffices to show
that there exist k > 0 and  2 L with 
  such that k  nWi=1
Udi,t_ μ6 B0 _ .
In fact, by assumption, for each di there exist ti > 0 and i 2 L with 
 i such that
kx kdi < ti for each x 2 M(LX) satisfying x
 B0 _ i, which implies that x
 Udi,ti
for each x 2 M(LX) satisfying x
 B0 _ i. Hence Udi,ti 6 B0 _ i.
Put tM = max{t1, t2, . . . , tn}, k = tM/t and  =
nWi=1
(i _ μ). Then k > 0, 
  and
k "  n _i=1
Udi,t!_ μ#=   n _i=1
Udi,tM!_ μ 6   n _i=1
Udi,ti!_ μ 6 B0 _ .
This completes the proof. 
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Corollary 5.1. Let (LX, {k · kd}d2D) be a locally convex L-tvs, where D = S 2M(L)
D.
Then B 2 LX is bounded iff for each  2 M(L) and d 2 D, there exist t > 0 and  2 L
with 
  such that kx kd < t for each x 2 M(LX) satisfying x
 B0 _ .
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