Advances in Pure Mathematics, 2012, 2, 41-44
http://dx.doi.org/10.4236/apm.2012.21010 Published Online January 2012 (http://www.SciRP.org/journal/apm)
L-Topological Spaces Based on Residuated Lattices
Zhudeng Wang1, Xuejun Liu2
1Department of Mathematics, Yancheng Teachers University, Yancheng, China
2School of Computer and Information Technology, Zhejiang Wanli University, Ningbo, China
Received September 14, 2011; revised October 10, 2011; accepted October 20, 2011
ABSTRACT
In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice
and discuss some properties of interior and left (right) closure operators.
Keywords: Residuated Lattice; L-Topological Space; Interior Operator; Left (Right) Closure Operator




1. Introduction
x
yzy xz .
Residuation is a fundamental concept of ordered struc-
tures and the residuated lattices, obtained by adding a
residuated monoid operation to lattices, have been applied
in several branches of mathematics, including L-groups,
ideal lattices of rings and multivalued logic. Commuta-
tive residuated lattices have been studied by Krull, Dil-
worth and Ward. These structures were generalized to the
non-commutative situation by Blount and Tsinakis [1].
Definition 1.1. [1-4]. A residuated lattice is an algebra
of type (2, 2, 2, 2, 2, 0, 0) sat-
isfying the following conditions:

,,,, ,,0,1LL



(L1) is a lattice,

,,L

,,1L(L2) is a monoid, i.e., is associative and
11
x
xx for any
x
L,
(L3)
x
yz if and only if
x
yz if and only if
y
xz,, for any
x
yz L.
Generally speaking, 1 is not the top element of L. A
residuated lattice with a constant 0 is called a pointed
residuated lattice or full Lambek algebra (FL-algebra, for
short). If 1
x
 for all
x
L, then L is called integral
residuated lattice. An FL-algebra L which satisfies the
condition 01
x
 for all
x
L

is called FLw-algebra
or bounded integral residuated lattice (see [2]). Clearly, if
L is an FLw-algebra, then


,,,0,1L

ab
11
is a bounded
lattice.
A bounded integral residuated lattice is called com-
mutative (see [5]) if the operation is commutative. We
adopt the usual convention of representing the monoid
operation by juxtaposition, writing ab for .
The following theorem collects some properties of
bounded integral residuated lattices (see [1-4,6].
Theorem 1.1. Let L be a bounded integral residuated
lattice. Then the following properties hold.
1) , 1xxxx
x
xx .
2)


,3)
x
xyxy


,


x
yx x y ,
x
yxy


.
y
xxy





,
x
yyz xz  4)




y
zxy xz
,. 5) If
x
y

then
x
,,zyzzxzy,
x

zy z
,
zyzz xzy .zxzy
and
x
6)
x
y

if and only if if and only if
1xy
1xy

.


,
x
yzy xzxyzxyz . 7)



 
,
x
yzxzyz 8)






.
x
yzxzyz  



 
,
x
yzxyx z 9)






x
yzxyx z .
If bounded integral residuated lattice L is complete,
then


,
x
zyLyxzxzyLxyz 

,, ,
jj
ababLjJ
Thus, it follows from some results in [7] that
Theorem 1.2. Let L be a complete bounded integral
residuated lattice and . Then the
following properties hold.


1)
j
JjjJ j
ab ab



,
jJ jjJ j
ab ab

 and
i.e., the operation

is infinitely -distributive. 




jJ jjJj
ab ab


and 2)




.
jJ jjJj
ab ab






3)
j
Jj jJj
ab ab

 and




j
Jj jJj
ab ab



, i.e., the two residuation
operations and are all right infinitely

-dis-
tributive (see [8]).




jJ jjJj
ab ab


and 4)




jJ jjJj
ab ab

.
C
opyright © 2012 SciRes. APM

Z. D. WANG ET AL.
42


5)
  
j
Jj
ab ,
X
L
jJ j
ab

 
 
.
and

j
Jj
ab
jJj
ab


Let us define on L two negations,
L
 and
R

:
0xx
,
L0xx

,
and .
R

For any j
x
xjJbL
, it follows from Theo-
rems 1.1 and 1.2 that
,
LR
x
x  ,
RL
x
x 

,
LL
x
yxy

,
RR
x
yxy ,
RL
x
yy x
,
LRL L
x
x  ,
RLR R
x
x 
RR
,
x
yy x  ,
LL
x
yy x 

,
LL
j
JjjJj
x
x 




,
RR
j
JjjJj
x
x

 

,
LL
j
JjjJ j
x
x

 


R. R
j
JjjJ j
x
x


RL

A bounded residuated lattice L is called an involutive
residuated lattice (see [3]) if LR
x
xx  for
any
x
L
,,
L
. In a complete involutive residuated lattice L,
RR L
x
yy x

,.
RR
xxyy 

LL
j
Jj jJjjJjjJj
x
xx x

 
.
X
L

,X
jLjJ
 
LR

 
In the sequel, unless otherwise stated, L always repre-
sents any given complete bounded integral residuated
lattice with maximal element 1 and minimal element 0.
The family of all L-fuzzy set in X will be denoted by
For any family of L-fuzzy sets,
we will write ,,
j
Jj

 
 and 
j
Jj



 

,,
R
to de-
note the L-fuzzy sets in X given by

  

LL


R
x
xxx  

 
   


,.
jjJjjJjjJjjJ
x
xxx 

 


Besides this, we define as follows:
1,0 X
XXL
1() 1
X
x
xX

00
X
and
x
xX

 .
2. L-Topological Spaces
A completely distributive lattice L is called a F-lattice, if
L has an order-reversing involution ':. When L is
a F-lattice, Liu and Luo [9] studied the concept of L-
topology. Below, we consider the notion of L-topological
space based on a complete bounded integral residuated
lattice.
LL
.
X
L  
Definition 2.1. Let If

satisfies the fol-
lowing three conditions:
(LFT1) 0,1
XX ,


(LFT2) ,,



  ,
(LFT3) jjJj



  

then

is called an L-topology on X and
When ][0,1L

, called an L-topological space

an F-topological space.

is called an open subset in .
X
L Every element in




L

,
X
L

L-
topological space.
Let L


R
 and

R


 . The
elements of
L


and
R


are called, respectively, left
closed subsets and right closed subsets in .
X
L

be an L-topology on X and Definition 2.2. Let

L-fuzzy subset of X. The interior, left closure and right
closure of

w.r.t

are, respectively, defined by




int ,



 




,
LL
cl




 


 
.
RR
cl




 
int,
L
cl and
R
cl are, respectively, called interior, left
closure and right closure operators.


int

, For the sake of convenience, we denote


,clL


R
cl o

and

by

,
L

, and
R


respec-
tively.
In view of Definitions 2.1 and 2.2, for any
,
X
L  


,
o








 
1
,
,,
LL
L
LL L


 
 
 


 
2
,
,,
RR
R
RR R





 
 
where


1,,
L






2,,
R




o
i.e.,

is just the largest open subset contained in ,

L


and
L


are, respectively, the smallest left closed
and right closed subsets containing .

For any
,
X
L  










,
,.
Lo LL
LL LL
L


 

 


 
 
 
.
RoR
Similarly,
R



,
X
L  

Lo L
Theorem 2.1. If L is an involutive residuated lattice
and then

1)
L

 
 
Ro R
and



R

 
oLR RL
;
2)
R
L




 
 
,,
oo
LLRR
;
3)
R
L



 

Z. D. WANG ET AL. 43

o
LR
L



.
o
RL
R and


 
 
Proof. When L is an involutive residuated lattice,
.
R
LLR
 
X
L  
X
L  
,
LL

1) If and

 

.
RL
then

RL
 
 

Lo L
 

. Thus,
L
 
 
R

 Similarly,
Ro
R




oRL
.
2) It follows from 1) that

o
 
,
RL
L
  
oLR


 
 
.
LR

o
R
  

o
LL


o
RR


o
LR


 
 
,
L L
R
R



 
,
RR
L
L





,
LR
3) By 2), we see that

R
 

L
 


LR
L
L



 


.
RL
 

o
RL



RL
R
R



 
 
0 0.
X X
LR


.
 
,

11,0
o
XXX

,,
o

X
L   Theorem 2.2. Let . Then the following
properties hold:
1)
2)
L
R

  
,



3) If

  then ,
oo

 ,
L
L



 .
R
R




4)

,
o
oo



L
L
L



 and

.
R
R
R





5)

.
ooo



 

LL
6) If ,,
L
x
yx
.
yxyL  then

L
L
L







RR
7) If ,,
R
x
yx
.
yxyL then

R
R
R






Proof. By Definition 2.2, it is easy to see that 1)-3)
hold.
4) By 2) and 3), we have that

.
o
oo

 On the
other hand, o

  and oo



. Thus, it follows from
Definition 2.1 that and so .

o
oo
  

o
oo



We can prove in an analogous way that
L
L
L




 and

.
R
R
R





5) Clearly,

.
ooo



 
,
oo Noting
that





.
o
oo oooo
we see that





  

.
ooo
Thus,



 
11
, 6) There exist



such that 1,
L
L



1
L
L


.
If


,,
LLL
x
yxyxyL

 

11
.
LLL
then
11
L
LL

 


 

Thus,
.
L
L
L







 Clearly,

.
L
L
L







Therefore, .
L
L
L








:XX
7) Similar to (6).
Theorem 2.3. Let
f
LL
X
L
be a mapping. Then
the following two statements hold.
1) If the operator f on satisfying the follwing
conditions:
(C1)


11,
XX
f
(C2)


,
X
f
L  
(C3)




,,
X
f
ff L  
then




,X
f
L   is an L-topology on X.
Moreover, if the operator f also fulfills
(C4)






,
X
f
ff L  
,
then with the L-topology


o
f


,
X
L  .
for every
i.e., f is the interior operator w.r.t

2) If the operator f on satisfying the follwing
conditions:
X
L
(C1')


00,
XX
f
(C2')


,
X
f
L  
(C3')




,,
X
f
ff L   then
a) when


,,
LLL
x
yxyxyL

 




1,
L
LX
f
L   
is an L-topology on X, moreover, if the operator f also
fulfills
(C4)






,
X
f
ff L   : and
L
XX
LL
is a bijection, then with the L-topology 1

,


,
X
L
f
L 

 i.e. , f is the left closure operator
w.r.t 1

;
b) when


,,
RRR
x
yxyxyL

 




2,
RR X
f
L   
:
is also an L-topology on X, moreover if (C4) holds and
R
XX
LL is a bijection, then with the L-topology
2

,


,
X
R
f
L 

 i.e., f is the right closure op-
erator w.r.t 2

.
Proof. 1) Refer to the proof of Theorem 2.2.2 1 in [9].
2) Clearly, 1
0,1.
XX


If 12 1 ,,

  then






 

121 2
1212
12
,
LLL
LLLL
L
ff
ff
 


 
 
 
121
i.e.,





1, jjJ
 . If then

Z. D. WANG ET AL.
44


.
LREFERENCES

 
LL
j
JjjJ j
LL
jJj
ff
jJ j
jJ j
f





 


.
LL
jJ j

 
 
[1] K. Blount and C. Tsinakis, “The Structure of Residuated
Lattices,” International Journal of Algebra and Compu-
tation, Vol. 13, No. 4, 2003, pp. 437-461.
Combing with (C2'), we have that [2] N. Galatos, P. Jipsen, T. Kowalski and H. One, “Residu-
ated Lattices: An Algebraic Glimpse at Substructural Lo-
gics,” Elsevier, Amsterdam, 2007.


jJ j
f



1jJ
 
Thus, j



 1 and so

is an L-topology on X.
For any
,
X
L  
 




 
[3] L. Z. Liu and K. T. Li, “Boolean Filters and Positive Im-
plicative Filters of Residuated Lattices,” Information Sci-
ences, Vol. 177, No. 24, 2007, pp. 5725-5738.
doi:10.1016/j.ins.2007.07.014
 
 
1
1
1
,
,
,,
LL
LL
L
LL
L
ff
f



 





 
 
 

 
 

.
[4] Z. D. Wang and J. X. Fang, “On v-Filters and Normal v-Filters of a Residuated Lattice with a Weak vt-Opera-
tor,” Information Sciences, Vol. 178, No. 17, 2008, pp.
3465-3473. doi:10.1016/j.ins.2008.04.003
[5] U. Hohle, “Commutative, Residuated L-Monoids,” In: U.
Hohle and E. P. Klement, Eds., Non-Classical Logics and
Their Applications to Fuzzy Subsets, Kluwer Academic
Publishers, Boston, Dordrecht, 1995, pp. 53-106.
i.e.,

L
L
ff


 Moreover, if (C4) holds and
:
L
X
L
X
L is a bijection, then
 



1
,.
X
LL
L
fL  f 


 
  

 
 

,
[6] A. M. Radzikowska and E. E. Kerre, “Fuzzy Rough Sets
Based on Residuated Lattices,” In: J. F. Peter et al., Eds.,
Transactions on Rough Sets II, LNCS 3135, 2004, pp.
278-296.
f Therefore,
L


 i.e., f is the left closure op-
erator w.r.t [7] Z. D. Wang and Y. D. Yu, “Pseudo-t-Norms and Implica-
tion Operators on a Complete Brouwerian Lattice,” Fuzzy
Sets and Systems, Vol. 132, No. 1, 2002, pp. 113-124.
doi:10.1016/S0165-0114(01)00210-X
1

.
We can prove in an analogous way that 2

is an L-
topology on X and the correspond ing f is the right closure
operator w.r.t [8] Z. D. Wang and J. X. Fang, “Residual Operations of Left
and Right Uninorms on a Complete Lattice,” Fuzzy Sets
and Systems, Vol. 160, No. 1, 2009, pp. 22-31.
doi:10.1016/j.fss.2008.03.001
2

.
3. Acknowledgements
This work is supported by Science Foundation of Yancheng
Teachers University (11YSYJB0201). [9] Y. M. Liu and M. K. Luo, “Fuzzy Topology,” World
Scientific Publishing, Singapore, 1997.