 Applied Mathematics, 2011, 2, 1522-1524 doi:10.4236/am.2011.212215 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Special Lattice of Rough Algebras Yonghong Liu School of Automation, Wuhan University of Technology, Wuhan, China E-mail: hylinin@163.com 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 . 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.  Let is a set. Define the meet and join ( operations by L())glb( ,),lub( ,).xyxyxyx y The following properties hold for all elements ,, .xyz L 1) commutative laws: and .xyyxxyyx  2) associative laws: ()() and () ().xy zx yzxy zx yz 3) absorption laws: () and ().xxyxx xyx  4) idempotent laws: and .xxx 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.  Let is a lattice, if for any L,, .xyz L 1) ()()(),or2)()() ().xyzxy xzxyzxy xz Therefore, is called a distributive lattice. LDefinition 1.3.  A distributive lattice is called a modular lattice. Theorem 1.1.  Let L is modular lattice, then L is called a semi-modular lattice. Definition 1.4.  Assume that is a finite and non-empty set with the universe, denote a binary relation on . Let is an approximation spaces. URUUU(,)URDefine (,)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.  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,.xyR If () ()Rx Ry, then the rough x and y Y. H. LIU1523 are called lower rough equal. The notation xy 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 xy denotes that x and y are upper rough equal. If xy and xy, then the rough x and y are called rough equal. The notation xy denotes that x and y are rough equal. Definition 2.2. Let , and be a unary operation. We use the notation ,ttxxS(),tttTttT tTxxS x ().ttttT tTtTxxS x  The definition represents that the rough union and rough intersection (where is an index set). TDefinition 2.3. Let algebra ,, be a rough lat-tice, if for any ,, ,xyz R satisfying ()(),xyxzy xzy   Therefore, is a rough modular lattice. RTheorem 2.1. The rough distributive lattice ,, is rough modular lattice. Proof. Suppose that is a rough distributive lattice, if for any R,, ,xyz Rxy, then ()(( ))(((()())(() ())()()()()()()().))xzyxyzSxyzxyxzSxyxzxyxz Sxyxzxyxz Sxyxzxy xzxz y      Hence the is rough modular lattice. ,,Theorem 2.2. The rough modular lattice is rough distributive lattice, if and only if for any ,,,, ,xyz R the following formulas hold: ()()() ()()(xyyz zx).xyyzzx Proof. Necessity: If for any ,,,xyz R then ()()() ((())(()))()xyyz zxxyy 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) ()()().xyyzzx Sufficiency: If for any ,, ,xyz R then () ()() ()()() x((xy)(yz)(zx)) (()()()) (()()()))xyzxxz yzxxy xzyzxxyyz zxxyyzzx x    Because ,xyx and since the rough modular laws. We see that (() (()())) ()((()())xyyz zxx).xyyzzxx  Because ,zxx and since the rough modular laws. It follows that ()()( )()().xyzxyzxxyxz We conclude that ()()().xyzxy 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,, xand ,yRxzRy  we have ,,xzyzxzyz then .xy Proof. Necessity: () () () () ()xx xzxyzxzyxz yyz yy.   Sufficiency: Let xy. To show that z, we thus have ()().xzyxzy  We shall prove that the two laws: 1) (() )( ()),2) (())(()).xzyzxzy zxzyzxzy z     To prove 1). In fact, (( ))(( )),(( )) (()), andxzyzxzzyzyxzy zyzy zyzzy    () (())yzzyyz zx zyz,   thus Copyright © 2011 SciRes. AM Y. H. LIU Copyright © 2011 SciRes. AM 1524 (()).xzyzzy For part 2), (( ))(( )).xzyzxzy zxz and (( ))().xzyzxzzxz  Since ,xy we conclude that ,xyx then () (()),xzxyz xzyz    thus (()),xzyzxz which proves 2). Definition 2.4. The is called a rough semi-modu- lar lattice denotes that Rx is coverage of xy, and y is also coverage of xy, then xy 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. RRProof. Let x be coverage of xy and let y is also coverage of xy, if for every fR, and ,xfxy which proves that fx, whence xy is coverage of x. In fact, ,xfxy we have () ,xyfyxyyy  but fyy, if not, xf and .yf This leads to the contradiction .xyf We see that y is cov-erage of xy, so that ,xyfy  thus ()()().ffyx fyxxyxx    Similarly, xy is coverage of y. Theorem 2.5. Assume that rough semi-modular lattice Then the Cartesian product is a rough semi-modular lattice. 12,,,.nRR RRRR12Proof. Let nR12(, ,,)nxxx x and 12(, , ,)nyyyy R, , and let 12 nRRR Rx, y are coverage of 12(, ,,)nxyzz z , if there exists , then iix is coverage of , and let , we have izkikzxk. If there exists j, then jy is coverage of zj, and let kj, we have kkzy. If ij, then 12(, ,, )nxytt t , where iitx, jjty, kktz (,kij). Here, xy is coverage of x, it is also coverage of y. If ij, then x12(, , , )nyu uu, where iiiuxy, kkuz (ki). Because i be a rough semi-modular lattice, hence i is coverage of Ruix, and it is also coverage of .iy Therefore, xy 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, ,XUR and be a set of equivalent relation, then is a rough semi-modular lattice based on the inclusion rela-tion. R 3. References  Z. Pawlak, “Rough Sets,” International Journal of Com-puter and Information Sciences, Vol. 11, No. 5, 1982, pp. 341-356. doi:10.1007/BF01001956  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.  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  Y. H. 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