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 E-mail: [email protected] 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 3) absorption laws: x (xy ) x and x ( xy ) x . It is generally known that the order and the partial order set theory were widely applied in the discrete mathemat- 4) idempotent laws: ics and fuzzy mathematics. In algebraic theories about x xxand xxx. the notion of lattice as both profound and sweeping. A lattice is an algebra structure L,, that has two The rough set was introduced by Pawlak in 1982 [1]. binary composition and , it satisfies the above- The lattice to characterize rough set is an important task mentioned condition 1), 2), 3) and 4). [2-6]. Actually, we can use a lattice model to represent Definition 1.2. [7] Let L is a lattice, if for any different information flow policies and play an important x,,yz L . role in Boolean algebra. Obviously, the coverage prob- 1) xyzxyxz ( ) ( ) ( ), or lem and the reductions problem are two problems of the cores in information system of lattice relation, which 2)xyzxyxz ( ) ( ) ( ). boosts the development of lattice theory. We give several Therefore, L is called a distributive lattice. special lattices of rough algebras that we discuss in this arti- Definition 1.3. [7] A distributive lattice is called a cle; for instance, we prove that a lattice is necessary and modular lattice. sufficient condition of the rough semi-modular lattice. Theorem 1.1. [7] Let L is modular lattice, then L is We will now describe a lattice definition and then we called a semi-modular lattice. introduce rough approximation spaces. The main con- Definition 1.4. [8] Assume that U is a finite and non- tents are as the following: empty set with the universe, RUU denote a binary Definition 1.1. [7] Let L is a set. Define the meet relation on U . Let (,)UR is an approximation spaces. () and join () operations by Define (,RR ) R U is a rough approximation x yxglb( ,y ),
x yxlub( ,y ). spaces. R and R are referred to as the lower and up- per approximation operators respectively. The following properties hold for all elements Theorem 1.2. [8] Let (,)UR be an approximation x,,yz L . spaces. Then algebra ,, is a complete distribu- 1) commutative laws: tive lattice. x yyx and xyyx . 2) associative laws: 2. Main Results
()xy zx () yz and Definition 2.1. Let (,)UR be an approximation spaces. ()xy zx (). yz For all x,.yR If Rx() Ry (), then the rough x and y
Copyright © 2011 SciRes. AM Y. H. LIU 1523 are called lower rough equal. The notation x y de- ()()()(yz xy zx xyz) notes that x and y are lower rough equal. If Rx() Ry (), (idempotent laws and commutative laws) then the rough x and y are called upper rough equal. The notation x y denotes that x and y are upper rough ()()(x yyzzx). equal. If x y and x y , then the rough x and y are Sufficiency: called rough equal. The notation x y denotes that x If for any x,,yz R , then and y are rough equal. xyzxxzyz () ()() Definition 2.2. Let xxtt, , and S be a unary operation. We use the notation x (xy ) ( xz ) ( yz ) x ((x y) (y z) (z x)) xttxSt (),x tT tT tT x ((xy ) ( yz ) ( zx )) xttxSt () x. tT tT tT ((x yyzzx ) ( ) ( ))) x The definition represents that the rough union and Because x yx , and since the rough modular laws. rough intersection (where T is an index set). We see that Definition 2.3. Let algebra ,, be a rough lat- ((xy ) (( yz ) ( zx ))) x tice, if for any x,,yz R , satisfying ()((()())x yyzzxx ). x yxzyxzy()() , Because zxx , and since the rough modular laws. Therefore, R is a rough modular lattice. It follows that Theorem 2.1. The rough distributive lattice ,, ()()(x yzxyzxxyxz )()(). is rough modular lattice. Proof. Suppose that R is a rough distributive lattice, We conclude that if for any x,,yz R ,x y , then x ()()(yz xy xz).
x ()(())((zy x yz Sxyz )) Show that ,, is the rough distributive lattice. ((x yxzSxyxz ) ( )) (( ) ( )) Theorem 2.3. A necessary and sufficient condition that rough lattice R is a rough modular lattice, for any ()()xyxz Sxyxz x,,yRx y and zR , we have
()()xyxz Sxyxz x zyzxzyz,, then x y. ()()xy xz Proof. Necessity: ().xz y xx () xz xyz () Hence the ,, is rough modular lattice. xzy () Theorem 2.2. The rough modular lattice ,, is rough distributive lattice, if and only if for any ()xz y x,,yz R , the following formulas hold: ()yz y y. ()()()xy yz zx Sufficiency:
()()(x yyzzx). Let x y . To show that z, we thus have Proof. Necessity: x ()()zy xz y. If for any x,,yz R , then We shall prove that the two laws: ()()()xy yz zx 1) (x (zy )) z (( xz ) y ) z ,
(((x yy ) ) (( xyz ) )) ( zx ) 2) (x (zy )) z (( xz ) y ) z . (distributive laws) To prove 1). In fact, ((yxzyz )())( zx) ((x zyz ) ) (( xzzyzy ) ) , (absorption laws and distributive laws) ((xzy )) z ()()(yz yx xzz )( xzx) ((yzyzyzzy )) , and
(y z z) (y z x) yzzy () yz z (()) x zy z, (distributive laws) thus
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((x zyzzy ) ) . i , then xi is coverage of zi , and let ki , we have zx . If there exists j, then y is coverage of z , and For part 2), kk j j let kj , we have z y. If ij , then ((x zy )) zx (()) zy z xz. kk x yttt(,12 , ,n ), where txii , tyj j , tzkk and ( kij , ). ((x zyzxzzxz ) ) ( ) . Here, x y is coverage of x , it is also coverage of y. If ij , then x yu(,12u,,) un , where Since x y, we conclude that x yx, then uxyii i, uzkk (k i). x zxyz() (()) xzyz , Because Ri be a rough semi-modular lattice, hence ui is coverage of xi , and it is also coverage of yi . thus Therefore, x y is not only coverage of x , but also ((x zyzxz ) ) , coverage of y . Corollary 2.1. Let (,UR) be an approximation which proves 2). spaces. Suppose that X is a nonempty set, X U, Definition 2.4. The R is called a rough semi-modu- and R be a set of equivalent relation, then R is a lar lattice denotes that x is coverage of x y , and y rough semi-modular lattice based on the inclusion rela- is also coverage of x y , then x y is coverage of tion. x and it is also coverage of y .
Theorem 2.4. If R is a rough modular lattice, then 3. References R is a rough semi-modular lattice. Proof. Let x be coverage of x y and let y is [1] Z. Pawlak, “Rough Sets,” International Journal of Com- also coverage of x y , if for every f R , and puter and Information Sciences, Vol. 11, No. 5, 1982, pp. x fxy, 341-356. doi:10.1007/BF01001956 [2] M. Novotny and Z. Pawlak, “Characterization of Rough which proves that f x , whence x y is coverage of Top Equalities and Rough Bottom Equalities,” Bulletin of x . In fact, the Polish Academy of Sciences. Mathematics, Vol. 33, x fxy, No. 1-2, 1985, pp. 91-97. [3] D. Dubois and H. Prade, “Rough Fuzzy Sets and Fuzzy we have Rough Sets,” International Journal of General Systems, x yfy() xyyy , Vol. 17, No. 2-3, 1990, pp. 191-209. doi:10.1080/03081079008935107 but f yy , if not, x f and y f . This leads [4] Y. H. Liu, “Lattice to Characterizing Rough Set,” Pattern to the contradiction x yf . We see that y is cov- Recognition and Artificial Intelligence, Vol. 16, No. 2, erage of x y , so that 2003, pp. 174-177. x yfy, [5] W. Q. Liu and C. X. Wu, “The Approximation Operator on F-Lattice,” Acta Mathematical Seneca, Vol. 46, No. 6, thus 2003, pp. 1163-1170. f fyxfyxxyxx()() () . [6] G. L. Liu, “The Lattice Properties of Rough Set Quotient Spaces,” Computer Engineering & Science, Vol. 26, No. Similarly, x y is coverage of y . 12, 2004, pp. 82-90. Theorem 2.5. Assume that rough semi-modular lattice [7] D. C. Sheng, “Abstract Algebra,” Science Press, Beijing, RR12,,,. Rn Then the Cartesian product 2001, pp. 114-127. RRR R is a rough semi-modular lattice. 12 n [8] W. X. Zhang, W. Z. Wu, J. Y. Liang and D. Y. Li, Proof. Let x (,xx12 , , xn ) and “Rough Sets: Theory and Methods,” Science Press, Bei- y (,yy12 , , yn ) R, RRR12 Rn , and let x , jing, 2001, pp. 72-77.
y are coverage of x yzzz(,12 , ,n ), if there exists
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