Rough Lattice: A Combination with the Lattice Theory and the Rough Set Theory

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1 06 Interntionl Conference on Mechtronics, Control nd Automtion Engineering (MCAE 06) ough Lttice: A Combintion with the Lttice Theory nd the ough Set Theory Yingcho Sho,*, Li Fu, Fei Ho 3 nd Keyun Qin 4 School of Informtion, Guizhou University of Finnce nd Economics, Guiyng, Guizhou, 55005, Chin School of Mthemtics & Sttistics, Qinghi Ntionlities University, Xining, Qinghi, 80007, Chin 3 Deprtment of Computer Softwre Engineering, Soonchunhyng University, Asn, 587,Kore 4 School of Mthemtics Southwest jiotong university Chengdu, Sichun, 6003, Chin * Corresponding uthor Abstrct The rough set theory, introduced by Pwlk in 98, is forml for deling with the uncertinties But it cnnot directly del with the uncertinties with order structure The lttice theory, introduced by Peirce nd Schr$\ddot{o}$der towrds the end of the nineteenth century, is mthemticl tool with order structure, lgebric structure nd topologicl structure In this pper, the rough theory is pplied to the lttice theory, nd the concept of the rough lttice is presented in order tht tool is presented which cn del with the uncertinties with lttice structure For this purpose, n equivlence reltion on lttice is defined nd then the notions of rough lttice nd lower nd upper pproximtions re introduced nd some relted properties re investigted At lstme relted lgebric structures re studied Keywords-rough set; lttice; rough lttice; lower pproximtion; upper pproximtion I INTODUCTION The theory of rough sets, proposed by Pwlk[0, is forml tool for modeling nd processing incomplete informtion in informtion systems It is n extension of set theory in which subset of universe is described by pir of ordinry sets clled lower nd upper pproximtions By using the concepts of lower nd upper pproximtions, knowledge hidden in informtion systems my be unrveled nd expressed in the form of decision rules The notion of rough set nd its properties hve been pplied to mny res by some reserchers Some importnt notions such s rough ring, rough hyperring, rough module nd rough group were rough module nd rough group were introduced[4,5,6,7,7,8 In [6, Xio nd Zhng introduced the notions of rough prime idels nd rough fuzzy prime idels in semigroup In[4, Jing et l studied the product structures of rough fuzzy sets on group nd proposed the notions of T-rough fuzzy subgroups in group with respect to T-fuzzy norml subgroup Ho et l[ Serched Miniml Attribute eduction Sets Bsed on Combintion of the Binry Discernibility Mtrix nd Grph Theory In [8,Xio nd Zhng introduced the notions of rough prime idels nd rough fuzzy prime idels in semigroup In [5, Kznc nd Dvvz further introduced the notions of rough prime(primry) idels nd rough fuzzy prime(primry) idels in ring nd presented some properties of such idels Now the rough set theory hs been successfully pplied to informtion reduction[3,6,30 The concept of lttice ws introduced by Peirce nd Schroder towrds the end of the nineteenth century It derived from pioneering work by Boole on the formliztion of propositionl logic[ In 930s Birkhoff's work strted the generl development of lttice theory[ Lttice re reltively simple structures since the bsic concepts of the theory include only orders, lest upper bounds, nd gretest lower bounds Now the lttice theory plys n importnt role in mny disciplines of computer science nd engineering For exmple, they hve pplictions in distributing, concurrency theory, progrmming lnguge semntics nd dt mining They re lso useful in other disciplines of mthemtics such s combintorics, number theory nd group theory [9,,8 Ho nd Zhong[3 studied the tg recommendtion bsed on user interest lttice mtching Presently, the work combining the lttice theory with the other mthemticl tools deling with uncertin informtion hs been initited For exmple, n nd oy[4 delt with rough set pproch on lttice theory Estji et l[8 introduced the notion of -upper nd -lower pproximtions of fuzzy subset of lttice Liu[7studied generlized rough sets over fuzzy lttices through both the constructive nd xiomtic pproches Estji et l[9 studied connection between rough sets nd lttice theory In this pper, they introduced the concepts of upper nd lower rough idels (filters) in lttice nd then offered some relted properties with regrd to prime idels(filters), the set of ll fixed points, compct elements nd homomorphisms Some reserchers even extended clssicl rough sets to Boolen lgebr[, completely distributive lttices[3 nd residuted lttices[3 Qin et l[begn the study of constructing the lttice structures of soft sets nd introduced the concept of soft equlity in 00 nd then the study of lttice structure hve become one of hot spots in relted fields[0,9,5 In this pper we pply the notion of rough sets to the lttice theory nd introduce the notion of rough lttice, which is different from tht presented by Estji etl[4,7,8,9, nd then derive some bsic properties nd discuss the lttice structure of rough idels (rough filters) of lttice The reminder of this pper is orgnized s follows Section presents some bsic concepts of the lttice theory nd the rough set theory Section 3 presents the notions of rough lttice nd derives some relted properties Section 4 concludes the pper 06 The uthors Published by Atlntis Press 009

2 06 Interntionl Conference on Mechtronics, Control nd Automtion Engineering (MCAE 06) II PELIMINAIES In this section the subjects needed in the other prts, hve been mentioned It is well known tht lttice is prtilly ordered set in which ny two elements x nd y hve lest upper bound x y nd gretest lower bound x y A sublttice ( L ;, ) of lttice ( L ;, ) is defined on non-empty subset L of L with the property tht, b L implies tht b, b L (, tken ( L ;, ) [,[ Next we give some importnt properties of nd Proposition:[, b, c L, ) nd ; ) b b nd b b ; If L is lttice, then for ll 3) ( b c ) ( b ) c nd ( b c) ( b) c ; 4) ( b) nd ( b) Proposition presents the chrcteristic properties of the opertions nd in the following theorem Theorem: [ Let L be nonempty set equipped with two binry opertions nd tht stisfy ()-(4) of Proposition If we define on L by b if nd only if b b, then ( L; ) is lttice in which the originl opertions gree with the induced ones, tht is, for ll b L b sup, b b inf, b,, nd Proposition revels the elegnt feture letting lttice be regrded either s ordered sets ( L ; ) or s lgebrs ( L ;, ) A nonempty subset I of L is clled lower set of L if for ll x L, there exists y I such tht x y implies x I ; dully, nonempty subset F of L is clled n upper set of L if for ll x L, there exists y F such tht y x implies x F Lemm [: ) If A nd B re two lower sets of L, then so re A B nd A B ) If A nd B re two upper sets of L, then so re A B nd A B The concepts of idel nd filter of lttice re usully used nd their properties re lso usully discussed A nonempty subset I of lttice L is clled n idel of L if it stisfies the follows: ) x y I, for ll, b I ; ) If x,then x I for ll I, x L Dully, nonempty subset F of lttice L is clled filter of L if it stisfies the follows: ) x y F,for ll, b F ; ) If x,then x F fir ll F, x L Theorem[9: Let I be nonempty subset of L, then the following ssertions re equivlent: ) I is n idel of L ; ) I is lower set nd I is closed under join; 3) x y L x y I x, y I,, dully, let F be nonempty subset of L, then the following ssertions re equivlent: ) F is filter of L ; ) F is n upper set nd F is closed under meet; 3) x y L x y F x, y F, In fct, n idel I of lttice L is lower set of L such tht x, y I implies x y I ; dully, filter F of lttice L is n upper set of L such tht x, y L implies x y F The following theorems, which re simple, re used often Theorem 3[, Let I nd I be two idels of L, then so is I I Dully, we hve the following theorem: Theorem 4[,: Let F nd F be two filter of L, then so is F F Definition [: Let L be lttice with lest element 0 nd gretest element An element is clled complemented element of L if there exists n element L such tht 0 nd is clled complement of Definition [: An element of L is clled distributive element of L if x y x y,for ll x, y L Theorem 5[: Let L be lttice, L, then is distributive element of L if nd only if for ll x, y L, it stisfies the followings, x y x y ); 06 The uthors Published by Atlntis Press 009

3 06 Interntionl Conference on Mechtronics, Control nd Automtion Engineering (MCAE 06) ) x y x y; 3) x y x y x ; 4) x y x y x Theorem 6[: Let be distributive element of L, then x y, x y implies x y An element of L is clled distributive complemented element if it is both distributive element nd complemented element of L It is cler tht ny lttice with lest element 0 nd gretest element hs t lest two distributive complemented elements: 0 nd In the rest of this section we recll some bsic definitions of set pproximtions Let U be non-empty set of objects clled the universe nd n equivlence reltion on U We use [ x to denote n equivlence clss in contining n element x U The pir ( U, ) is clled n pproximtion spce For ny X U, we cn define the lower nd upper pproximtion of X by _( X ) x [ X X, X ) x [ x X (, respectively The pir ( _( X ), ( X )) is referred to s the rough set of X The rough set ( _( X ), ( X )) gives rise to description of X under the present knowledge, ie, the clssifiction of U The following proposition is well known nd esily seen Proposition [0: Let ( U, ) be n pproximtion spce For every subsets X, Y U, we hve ) _( X ) X ( X ) ; ) ( X Y ) ( X ) ( Y ), ( X Y ) ( X ) ( Y ) ; 3) _( X ) _( Y ) _( X Y ), _( X Y ) _( X ) _( Y ) By mens of the bove concepts nd opertions of rough sets, we cn propose the concepts of rough lttices nd study some relted properties III OUGH LATTICES Let L be lttice with lest element 0 nd gretest element nd mpping L defined s follow: ( x) x for ll x L, where is distributive complemented element of L It is trivil to verify tht is lttice homomorphism is binry reltion on L by setting: x y if nd only if ( x) ( y), tht is, x y, x, y L It is obvious tht is n equivlence reltion on L Throughout this pper, L is lttice with lest element 0 nd gretest element, is distributive complemented element of L, is mpping on L, nd is n equivlence reltion on L s defined in the bove unless otherwise specified For ll L, the equivlence clss of L is the set x ( x) ( ) We denote it by [ By the definitions of upper nd lower pproximtions, we hve A) x[ x A ( A) x[ x A _(, Definition 3: Let A L A is clled n upper rough sublttice of L if (A) is sublttice of L Similrly, A is clled lower rough sublttice of L if (A) is sublttice of L A subset of L is clled rough sublttice of L if it is both n upper rough sublttice nd lower rough sublttice For convenience, rough sublttice of L is clled rough lttice of L Similrly, subset of L is clled rough idel of L if its upper pproximtion nd lower pproximtion re both n idel of L ; dully, subset of L is clled rough filter of L if its upper pproximtion nd lower pproximtion re both filter of L Proposition 3 [ is sublttice of L for ll L Proof x [, x [, we hve ( x ) ( ), ( x ) ( ) Since is lttice homomorphism, ( x x) ( x) ( x) ( ) ( ) ( ), ( x x) ( x) ( x) ( ) ( ) ( ),so x x [, x x [ Thus, [ is sublttice of L for ll L Suppose tht x A B x b, A, b B, A B x x b, A, b B In generl speking, A A A, A A A don't hold, but we hve the following: Lemm A is sublttice of L if nd only if A A A, A A A Proof: If A is sublttice of L, then we hve x x x for ll x A x A A, therefore, 06 The uthors Published by Atlntis Press 0093

4 06 Interntionl Conference on Mechtronics, Control nd Automtion Engineering (MCAE 06) A A A On the other hnd, for ll x A A, there exist, b A such tht x b A Since A is sublttice of L, we cn get tht x b A, therefore, A A A Thus, A A A Similrly, we cn get tht A A A If A A A, A A A, then for ll, b A, we cn get tht b A A A nd b A A A, tht is, b A, b A A is sublttice of L The following lemm helps us to derive some importnt properties of the rough lttices Lemm 3: For ll x L, L, L, ) [ [ ; ) x [ if nd only if x [ Proof : ) It cn be get immeditely by ( ) ( ) ) Since x [, we hve x x) ( ) x x is, x [ (,tht Since x [,we hve ( x), so x ( ) x ( x),tht is, x [ Proposition 4: For ll ) [ L, L, ; [ [ ) [ [ [ Proof: ) For ll x [,we hve x ( ) Let y ( x ( )) ( x ( )) On one hnd,since is distributive element of L, we hve y [( x ( )) ( x ( )) [( x) ( ) [( x) ( ) [( ) ( ) [( ) ( ) ( ) ( ) x, nd y [( x ( )) ( x ( )) [ x ( ) [ x ( ) ( x) ( x) x we cn get tht x y On the other hnd, since ( x ( )) ( x) ( ) ( ) ( ) [ Thus, we cn get tht [ ll Secondly, for [ [ x [ [, we cn get tht there exist [, [ such tht x x ( ) [ [ [ x [, tht is, Sum up bove, we cn get tht [ [ [ Let ) For ll x [, we hve ( ) x y [( x) ( ) [( x) ( ),where, is the complement of On one hnd, since is distributive complemented element of L, we hve y ([( x) ( ) [( x) ( )) ( [( x) ( )) ( [( x) ( )) ( ( )) ( ( )) ( ) ( ) ( ) x, nd y ([( x) ( ) [( x) ( )) ( [( x) ( )) ( [( x) ( )) [( ( x)) ( ( )) [( ( x)) ( ( )) ( x) ( x) x,so we cn get tht x y On the other hnd, since [( x) ( ) [ ( x) [ ( ) [( ) ( ) [,nd [( x) ( ) [ ( x) [ ( ) [( ) ( ) [,so we hve y [ [, which implies x [ [ Thus, we cn get tht Secondly, for ll x [ [ [ [ [ [, there exist, [ such tht, nd x x ( ) ( ) ( ) ( ) ( ) ( ), tht is, x [ Thus, we hve[ [ [ Sum up bove, we cn get tht [ [ [ 06 The uthors Published by Atlntis Press 0094

5 06 Interntionl Conference on Mechtronics, Control nd Automtion Engineering (MCAE 06) By the proof of Proposition 4, we cn get tht Proposition 4() is still true even if is not complemented element of L IV CONCLUSION We hve proposed the concepts of rough lttices nd studied their bsic properties Although mny results reported here re only concerned with bsic properties bout these new notions, they cn hopefully provide more insight into nd full understnding of rough sets theory nd one could see tht this study presents very preliminry, but potentilly interesting reserch direction Nturlly, very importnt nd interesting problem is how to pply the models given in this pper to build the corresponding theory of knowledge discovery in lttice-vlued informtion systems ACKNOWLEDGMENT The uthors re indebted to Professor F Feng for helpful discussion nd vluble suggestions This work ws supported by Qinghi Science Foundtion(Grnt No03-Z-93) nd Science nd Technology Fund Project of Guizhou Province(Grnt NoLKB[00) EFEENCES [ NAjml nd AJin, "Some constructions of the join of fuzzy subgroups nd certin lttices of fuzzy subgroups with supproperty", Informtion Sciences,vol,79,pp ,995 [ GBirkhoff, \emph{lttice Theory}, Trns Amer Mth Soc, New York, 967 [3 DChen, WZhng, DYeung nd ECC Tsng, "ough pproximtions on complete completely distributive lttice with pplictions to generlized rough sets", Informtion Sciences,vol,76, pp89-848,006 [4 BDvvz, "oughness in rings", Informtion Sciences, vol, 64, pp47-63,004 [5 BDvvz, "oughness bsed on fuzzy idels",informtion Sciences,vol, 76, pp47-437,006 [6 BDvvz, Approximtions in hyperrings, Journl of Multiple- Vlued Logic nd Soft computing 5(009) [7 BDvvz, MMhdvipour, oughness in modules, Informtion Sciences,vol, 76, pp ,006 [8 AAEstji, SKhodii nd SBhrmi, On rough set nd fuzzy sublttice, Informtion Sciences,vol, 8,pp ,0 [9 AAEstji, MHooshmndsl nd BDvvz, ough set theory pplied to lttice theory, Informtion Sciences, vol,00,pp08-,0 [0 L Fu, Soft Lttices, Globl Journl of Science Frontier eserch Vol,0(4), pp57-59,00 [ GGrtzer, Generl Lttice Theory, Acdemic Press, INC 978 [ FHo, ZPei nd SZhong, Serching Miniml Attribute eduction Sets Bsed on Combintion of the Binry Discernibility Mtrix nd Grph Theory, Interntionl Conference on Fuzzy Systems (FUZZ- IEEE 008), pp54-57 [3 FHo nd SZhong, Tg ecommendtion Bsed on User Interest Lttice Mtching, Interntionl Conference on Computer Science nd Informtion Technology (ICCSIT00), pp76-80 [4 JJing, CWu nd D Chen, The product structure of fuzzy rough sets on group nd the rough T-fuzzy Group, Informtion Science, vol,75, pp97-07,005 [5 OKznc nd BDvvz, On the structure of rough prime(primry) idels nd rough fuzzy prime(primry)idels in commuttive ring, Informtion Sciences,vol,78,pp ,008 [6 [NKuroki nd PPWng, The lower nd upper pproximtions in fuzzy Group, Informtion Sciences, vol,90,pp03-0,996 [7 GLiu, Generlized rough sets over fuzzy lttices, Informtion Sciences, vol,78,pp65-66,008 [8 YLi, Finite utomt theory with membership vlues in lttices, Inform Sci, vol,8(5),pp003-07,0 [9 MMrudi nd Vjendrn, New Construction of Fuzzy Soft Lttices,Interntionl Journl of Computer Applictions, vol,3(), pp33-38,0 [0 Z Pwlk, ough sets, Interntionl Journl of Computer nd Informtion Sciences,vol,,pp34-356,98 [ GQi nd WLiu, ough opertions on Boolen lgebrs, Informtion Sciences,vol,73,pp49-63,00 [ KQin nd ZHong, On soft equlity, JComputApplMth Vol,34,pp ,00 [3 AMdzikowsk nd EEKerre, Fuzzy rough sets bsed on residuted lttices, Trnsctions on ough Sets, Lecture Notes in Computer Sciences, vol,335,pp78-96,004 [4 Dn nd SKoy, ough Set Approch on Lttice, Journl of Uncertin Systems, vol,5(),pp7-80,0 [5 YSho nd KQin, Fuzzy Soft Sets nd Fuzzy Soft Lttices, Interntionl Journl of Computtionl Intelligence Systems, vol,5(6),pp 35-47,0 [6 DSlezk nd WZirko, The investigtion of the Byesin rough set Model, Interntionl Journl of Approximte esoning, vol, 40,pp8-9,005 [7 CWng nd DChen, A short note on some properties of rough groups, Computers nd Mthemtics with Applictions, vol, 59,pp43-436,00 [8 QXio nd ZZhng, ough prime idels nd rough fuzzy prime idels in Semigroups, Informtion Sciences,vol,76,pp75-733,006 [9 X Zhng, Fuzzy logic nd its lgebric nlysis, Science Press 008(In Chinese) [30 WZhu nd FWng, eduction nd xiomiztion of covering generlized rough Sets, Informtion Sciences vol,5,pp 7-30, The uthors Published by Atlntis Press 0095