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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
Email: [email protected], [email protected]
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) ,
1xxxx
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
xxx






 

,.
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

 
Copyright © 2012 SciRes. APM
Z. D. WANG ET AL. 43
Copyright © 2012 SciRes. APM

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.
Copyright © 2012 SciRes. APM
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.