Applied Mathematics, 2011, 2, 1522-1524
doi:10.4236/am.2011.212215 Published Online December 2011 (http://www.SciRP.org/journal/am)
Special Lattice of Rough Algebras
Yonghong Liu
School of Automation, Wuhan University of Technology, Wuhan, China
Received October 14, 2011; revised November 16, 2011; accepted November 24, 2011
Abstract
This paper deals with the study of the special lattices of rough algebras. We discussed the basic properties
such as the rough distributive lattice; the rough modular lattice and the rough semi-modular lattice etc., some
results of lattice are generalized in this paper. The modular lattice of rough algebraic structure can provide
academic base and proofs to analyze the coverage question and the reduction question in information system.
Keywords: Lattice, Rough Distributive Lattice, Rough Modular Lattice, Rough Semi-Modular Lattice
1. Introduction
It is generally known that the order and the partial order
set theory were widely applied in the discrete mathemat-
ics and fuzzy mathematics. In algebraic theories about
the notion of lattice as both profound and sweeping.
The rough set was introduced by Pawlak in 1982 [1].
The lattice to characterize rough set is an important task
[2-6]. Actually, we can use a lattice model to represent
different information flow policies and play an important
role in Boolean algebra. Obviously, the coverage prob-
lem and the reductions problem are two problems of the
cores in information system of lattice relation, which
boosts the development of lattice theory. We give several
special lattices of rough algebras that we discuss in this arti-
cle; for instance, we prove that a lattice is necessary and
sufficient condition of the rough semi-modular lattice.
We will now describe a lattice definition and then we
introduce rough approximation spaces. The main con-
tents are as the following:
Definition 1.1. [7] Let is a set. Define the meet
and join ( operations by
L
())
glb( ,),
lub( ,).
x
yxy
x
yx

 y
The following properties hold for all elements
,, .
x
yz L
1) commutative laws:
and .
x
yyxxyyx 
2) associative laws:
()() and
() ().
xy zx yz
xy zx yz


3) absorption laws:
() and ().
x
xyxx xyx

 
4) idempotent laws:
and .
x
xx xxx


A lattice is an algebra structure that
has two
binary composition ,,L

and , it satisfies the above-
mentioned condition 1 ), 2), 3) and 4) .
Definition 1.2. [7] Let is a lattice, if for any L
,, .
x
yz L

1) ()()(),or
2)()() ().
xyzxy xz
xyzxy xz



Therefore, is called a distributive lattice.
L
Definition 1.3. [7] A distributive lattice is called a
modular lattice.
Theorem 1.1. [7] Let L is modular lattice, then L is
called a semi-modular lattice.
Definition 1.4. [8] Assume that is a finite and non-
empty set with the universe, denote a binary
relation on . Let is an approximation spaces.
U
RUU
U(,)UR
Define


(,)RR RU is a rough approximation
spaces. R and R are referred to as the lower and up-
per approximation oper ators respectively.
Theorem 1.2. [8] Let be an approximation
spaces. Then algebra (,)UR ,,

 is a complete distribu-
tive lattice.
2. Main Results
Definition 2.1. Let be an approximation spaces.
For all (,)UR ,.
x
yR

If () ()Rx Ry

, then the rough x and y

Y. H. LIU1523
are called lower rough equal. The notation
x
y de-
notes that x and y are lower rough equal. If () ()Rx Ry,
then the rough x and y are called upper rough equal. The
notation
x
y denotes that x and y are upper rough
equal. If
x
y and
x
y, then the rough x and y are
called rough equal. The notation
x
y

denotes that x
and y are rough equal.
Definition 2.2. Let , and be a unary
operation. We use the notation
,
tt
xxS
(),
tt
tT
t
tT tT
x
xS

 x


().
tt t
tT tTtT
x
xS x
 

The definition represents that the rough union and
rough intersection (where is an index set). T
Definition 2.3. Let algebra ,,

 be a rough lat-
tice, if for any ,, ,
x
yz R satisfying
()(),
x
yxzy xzy  
Therefore, is a rough modular lattice. R
Theorem 2.1. The rough distributive lattice ,,


is rough modular lattice.
Proof. Suppose that is a rough distributive lattice,
if for any R
,, ,
x
yz R
x
y, then
()(( ))((
(()())(() ())
()()
()()
()()
().
))
x
zyxyzSxyz
x
yxzSxyxz
xyxz Sxyxz
xyxz Sxyxz
xy xz
xz y
   

 
 


Hence the is rough modular lattice. ,,
Theorem 2.2. The rough modular lattice is
rough distributive lattice, if and only if for any ,,
,, ,
x
yz R the following formulas hold:
()()()
()()(
xyyz zx
).
x
yyzzx


Proof. Necessity:
If for any ,,,
x
yz R then
()()()
((())(()))()
xyyz zx
x
yy xyz zx


(distributive laws)
(( )())(yxz yzzx )
)
)
(absorption laws and distributive laws)
()()( )(
(yzz)(yzx)
yzyx xzz xzx

(distributive laws)
()()()(yz xy zxxyz


(idempotent laws and commutative laws)
()()().
x
yyzzx


Sufficiency:
If for any ,, ,
x
yz R

then
() ()()
()()()
x((xy)(yz)(zx))
(()()())
(()()()))
xyzxxz yz
x
xy xzyz
x
xyyz zx
x
yyzzx x


 
 


 
Because ,
x
yx

 and since the rough modular laws.
We see that
(() (()()))
()((()())
xyyz zxx
).
x
yyzzxx

 

Because ,zxx

 and since the rough modular laws.
It follows that
()()( )()().
x
yzxyzxxyxz


We conclude that
()()().
x
yzxy xz


Show that ,,

 is the rough distributive lattice.
Theorem 2.3. A necessary and sufficient condition that
rough lattice is a rough modular lattice, for any R
,,
x
and ,yRxzRy  we have

,,
x
zyzxzyz then .
x
y
Proof. Necessity: ()
()
()
()
()
xx xz
xyz
xzy
xz y
yz yy.


 
 



Sufficiency:
Let
x
y. To show that z, we thus have
()().
x
zyxzy 
We shall prove that the two laws:
1) (() )( ()),
2) (())(()).
x
zyzxzy z
x
zyzxzy z
  
  
To prove 1). In fact,
(( ))(( )),
(( ))
(()), and
x
zyzxzzyzy
xzy z
yzy zyzzy
 
 
  
() (())yzzyyz zx zyz,

  
thus

Y. H. LIU 1524
(()).
x
zyzzy
For part 2),
(( ))(( )).
x
zyzxzy zxz
and
(( ))().
x
zyzxzzxz 
Since ,
x
y we conclude that ,
x
yx then
() (()),
x
zxyz xzyz   
thus
(()),
x
zyzxz
which proves 2).
Definition 2.4. The is called a rough semi-modu-
lar lattice denotes that R
x
is coverage of
x
y, and
y
is also coverage of
x
y, then
x
y is coverage of
x
and it is also coverage of
y
.
Theorem 2.4. If is a rough modular lattice, then
is a rough semi-modular lattice.
R
RProof. Let
x
be coverage of
x
y and let
y
is
also coverage of
x
y, if for every
f
R, and
,
x
fxy
which proves that
f
x, whence
x
y is coverage of
x
. In fact, ,
x
fxy
we have
() ,
x
yfyxyyy 
but
f
yy

, if not,
x
f and .
y
f This leads
to the contradiction .
x
yf We see that
y
is cov-
erage of
x
y, so that
,
x
yfy 
thus
()()().
f
fyx fyxxyxx   
Similarly,
x
y is coverage of
y
.
Theorem 2.5. Assume that rough semi-modular lattice
Then the Cartesian product
is a rough semi-modular lattice.
12
,,,.
n
RR R
RRR
12
Proof. Let n
R
12
(, ,,)
n
x
xx x and
12
(, , ,)
n
y
yyy R, , and let
12 n
RRR R
x
,
y
are coverage of 12 (, ,,)
n
x
yzz z , if there exists
, then
ii
x
is coverage of , and let , we have
i
zki

k
zx
k

. If there exists j, then
j
y
is coverage of zj, and
let kj

, we have kk
zy
. If ij

, then

12
(, ,, )
n
x
ytt t , where ii
tx,
j
j
ty, kk
tz

(,kij

).
Here,
x
y is coverage of
x
, it is also coverage of
y. If ij

, then
x
12
(, , , )
n
yu uu, where
iii
uxy

, kk
uz

(ki

).
Because i be a rough semi-modular lattice, hence
i is coverage of
R
ui
x
, and it is also coverage of .
i
y
Therefore,
x
y is not only coverage of
x
, but also
coverage of
y
.
Corollary 2.1. Let be an approximation
spaces. Suppose that (,UR)
X
is a nonempty set, ,
X
U
R and be a set of equivalent relation, then is a
rough semi-modular lattice based on the inclusion rela-
tion.
R
3. References
[1] Z. Pawlak, “Rough Sets,” International Journal of Com-
puter and Information Sciences, Vol. 11, No. 5, 1982, pp.
341-356. doi:10.1007/BF01001956
[2] M. Novotny and Z. Pawlak, “Characterization of Rough
Top Equalities and Rough Bottom Equalities,” Bulletin of
the Polish Academy of Sciences. Mathematics, Vol. 33,
No. 1-2, 1985, pp. 91-97.
[3] D. Dubois and H. Prade, “Rough Fuzzy Sets and Fuzzy
Rough Sets,” International Journal of General Systems,
Vol. 17, No. 2-3, 1990, pp. 191-209.
doi:10.1080/03081079008935107
[4] Y. H. Liu, “Lattice to Characterizing Rough Set,” Pattern
Recognition and Artificial Intelligence, Vol. 16, No. 2,
2003, pp. 174-177.
[5] W. Q. Liu and C. X. Wu, “The Approximation Operator
on F-Lattice,” Acta Mathematical Seneca, Vol. 46, No. 6,
2003, pp. 1163-1170.
[6] G. L. Liu, “The Lattice Properties of Rough Set Quotient
Spaces,” Computer Engineering & Science, Vol. 26, No.
12, 2004, pp. 82-90.
[7] D. C. Sheng, “Abstract Algebra,” Science Press, Beijing,
2001, pp. 114-127.
[8] W. X. Zhang, W. Z. Wu, J. Y. Liang and D. Y. Li,
“Rough Sets: Theory and Methods,” Science Press, Bei-
jing, 2001, pp. 72-77.