V N U J O U R N A L OF SCIENCE Nat Sci., t XV n^5 - 1999 C O N T R A C T IB L E SU BL A TT IC E S IN DATA A NALYSIS N guyen D ue D at Fnciiity o f Mathematics, Mecbãiiics Hiid ỉììỉonììHtỉcs College o f NHtìirãỉ Sciences - V N U IN T R O D U C T IO N Nowadays, the applying of lattice theory to concept analysis and d a ta analysis is of great interests In [6] R Wiile has proposed the notion of concept lattice Concerning the co nstn u tion of this lattice-, some prohlenis as siihdiiect (leconiposition, tensorial decomposition have been studied ill [7, Fiirther, the notion of concept lattice is widely used ill d a ta analysis, for example, in [3 4, 5, 9] P Luksch and R Wille [4] have proposed a decomposition of a context (O, o R) into subcontexts, which are iiidocomposablo In this paper, wo stu d y contiactihle sub latticrs [1] of a concept lattice B ( A, R) and with its help, we propose a ciccoinposition of context (O, A R) into pairwisely disjoint subcontexts and its quotient contf'xt (letpnninod by tliesp svihcontexts C O N C E P T S AND RESULTS First, rocall sonio noiions from R VVillo [6 D e f in i ti o n 2.1 B y tiic SVỈIÌÌ)0 Ì (O A R) we dcỉiote H context.' where o A aie Hibitrni y '.('is Ỉìiìỉl ỊÌ is H Ììiĩỉỉìrv rcỉĩìtioii hrtwiH'ii o Hiid A llio oìoìììeỉìts o f o Hiưì A a/e called ()i)jects riiidiìttriỉìììtcs rcspcctivelv- If ììtea I f O' c o A' c A ỉìỊid /?' = /? n O' X A' then ( \ A \ R ')is cnllcd ri si//>coỉJíeA'í (O A R) The relation R establishes a Gaiois connection between the power sets of o and A as follows; X* = {a e A\{.r,a) e R v.r € A'} f o r X c o, y * ^ {v e \ { v y ) e R Vy G Y } f o r Y c >1 D e f i n i t i o n 2.2 A concept o f the context (O, A, R) is defined as a pair (M, N), where M c O N c A, Sĩich t h a t A/* — N ãìid N* = M The fninily o f ỉỉll concepts o f (O, A, R ) ate deỉioted hy B ( , A, i?j On B ( , A R ) are defined relation < Hiid the Ịãttice o p e r a ti o n s A, V as follows: Nguyen D ue D at (H) < (A/ , ^ ) i f A/i c M (b) A,g/(A/,:, A^,).= (n,gyA /,, (n,e/A /,)*) (c) V i g / ( M j , A /,) = ( ( n , g / A ^ i ) * , n , g y A^,) It is caiiy to dom onstrate th at B ( A, R) is a roiuplete lattice [6 D e f i n i ti o n 2.3 Lattice B ( , A, R ) is called a c o n ce pt la tt ic e o f the context (O, A, R) Ill this paper wo consider the set of objocts and set of attributes, which are the objects are denoted by a, b, c etc , the attrib u tes by 1, 2, etc E x a m p l e Consider the contexts C] = [ , A , R \ ) and C = [ O A , R ) (F ig l) whcic o = {nJ)A',d}, A = { ,2 ,3 ,4 } These contexts deterniine the concept lattices Bi = B { O A R i ) and Bo = B { O A , R ) respectively For denoting a concept, for exam ple { M N ) e By with M = { a , b , r ì } , N = {1,3}, we shall write (abd, 13) instead of {{a.b.d},{h3\) 3 a X 'x X a X b X X b X X c X X X c X X X d X X X c X X X X X X Fig Now we shall deal with tilt' concepts of contractiblp sublatticp [1 D e f i n i ti o n 2.4 Let L be a lattice ) A siihlattice c o f L is called convex i f a < :r < h with n.b e c then T e c 2) I f n,h e L , is iincoinparable with b mid {c, d} = {a A Ò, rt V b},c Ỷ suhÌHttice then b,c,(i} is cnlled a square o f L and it is denoted by < a ,h ;c ,d > 3) A p i o p e i Sìiììỉattice c o f L wit h |C| > is called a contiHCtihle sii hl ntt ic e i f satisfies the following cuiiditioĩis: c C o n tr a c t i b le auhlattices in d a ta a n a ly sis ‘3 a) c is convex b) I f < a, h] (\ d > i.s a squárc Ji] L then r G c f/G c In [2] thoiP has aheaclv been proved: P r o p o s i t i o n Let c he a contiHctìì)ìe siihlãttỉce of a lãttice L and k e L \ C , c e c then: {Pi ) If k < c then k < :r, v.r E c {Pyj If k > c then k > ;r,v.r G c (P;í) I f k is ìincuinpHrHỈ)le with c then k IS ìincoiiipíìrãbìe w it h :r,V.T e c \ D e f i n i t i o n 2.6 We say that latticc L has a linear decomposition (or is linễilv decojuposỉìỉAe) if th er e e x i s t a chain I with |/ | > a n d sìih ìattices L,;, Ĩ e I Silch t h a t L = nnd for i , j e L ĩ < J theji a < b, Vi'; L,, V6 Lj L e m m a 2.7 Le t C x C ) }>e C0 iitỉ'HCtiì)le sìihlãttĩces o f L such that one d o e s n o t contain the otỉìei and Cl n Co / 0, then C\ u C -2 is a linearly decomposa.ì)ỉe subìãttice Fiirther, i f C \ u C -2 / L tíìCỉì i t is H c o ĩ i t i a c t i h ỉ e S ì ỉ h l a t t i c e Proof Using t h r properties Ỉ P ị ) , { P ) A P ị ) we come to the conclution of the le i n u i a ộ L e m m a Let {(T';|/ e /} Ỉ)CH family o f contractibie sìỉỉìlattices o f L such thiit C , n C j (Ỉ) V/ / e 1,1 ^ J Tiivii on L there exists a coij^rfjeijce Silch th at e v e r y C\^ e / , — is HU OiỊìiiVỉiìvnt cbiss niicl tỉiC otiiers Hie one - elemeiiĩ cÌHsses Proof, a) iletino an equivalence f) oil I which has the classes as c , V G I and {.r} r e Ả \ u e / C 1)) L(‘t (I ỊÌ (Ỉand Ỉ)Ị) b , we have to prove that (a A h)p{a (' I)*’" or ti.ii.d Ji r c \ foi noiiìc' i c / , it iu Here To = (f/ e 4) /, i/o = (e / , 6) 5) Ti = { c d e f , 0124) y, = {e / , T2 = (« d e / , 3) U2 (e/.r, 5) ^3 — {a b c d e f r, r,3 = (a c (I e / , 2) 0) Xị = {n b d e f , 13) Ttj = (a h c cl e f 1) In B ( , yl /Ỉ) then^ exists a contracti])le sublattice c = [:ro, X‘5 ] (interval) By (2.8) there exists a congiueiico f) on D { , A, R) : (t = (ipb I: U1 Thus, we have a quotient lattice B { , A, H ) / p (Fig 2c) We waìiĩ to consti n e t with a help o f tho context c \ a context defining s ubl attice c and another coiit('xt (h'tiiuu^ B {O A R )/p a) P u ttin g o = {aj),c,d}, A — { ,2 ,3 ,4 } and R — R n O x A , we have a now context {O , A , R ), which is a subcontpxt of c (given by a small square in Fig.2a) It is easy to S(*e th a t c ^ B { , A R ) (see also the context C] and coiici'pt lattice B\ in Fig.l) b) P u ttin g and basing c we construct a context - { ,/l',5 ,6 } : A’ X X 0* X X e X X X f X X X X X C o n tr a c t i b le su b la ttice s in d ata a naly sis B { , A , R)/() (see a k o the This context has a concept lattifc isomorphic to context C ‘2 and concept lattice ill Fig 0- Cont.oxt ( context of by Silhcontoxt c\ ^ , is calli'd a quotient Now, vvroposed notion siippralteniative for a conU'xt (O o , R) (here o — 4) In this paper, we define ''sup eralteinativ e’" for (O A, /?) D e fin itio n 2.11 Le t \ O Hijii A \ , A , A -2 are th e p r w is el y disjoint siilj s e t s in o rijid A, respectively, such thỉìt o — Oi u o u Ơ-2 , A = A \ u Ấ u A We say that H pair ỈO' A ) is a pi-iir o f siipcniltcrnativcs ÌỈÌ ( ,.4 , R) if: N o t e 2.12 ■/' (-){ tỉi ỉi (.r,o) G i? Vo G Ả ;r € O tỉien (.r,a ) Ệ R, Va e A a € A\ tiien (:i\a) G /?, V:í’ e a G Ao tlivn (.r,a ) ị R, Vx G o ■Ĩ' G *4] tije/j (r, it) G ỈỈ o I f (O' , A') is H p a i l o f s ii p e i a h e n i a t i v e s , pìỉttỉiig R' — R n O ' X A \ we have >i c o i itc x t { O ' , A \ IV) which is H Siihcontext o f (O, /?) P r o p o s i t i o n 2.13 I f C' is H c u n t i H C t i b l e s i i l j l n t t i c c o f B { /1 ỈỈ) t h e n t h c i v e x i s t s H pair of siipcj nltcj luitiws {O' A') siicij t iia t c ^ B ( \ A \ R^) Proof Suppos(' that (A/,, N,) i / are all eltMiierits of c P u t M q — N{ n,, /.V/, Ao — A/q, M \ = A'f then (A/(1 'Vo) and ( M \ N \) are the smallest and greatest ' e r :,,a ‘ L S) and u \V(“ Ị)ut u — M\ \ h \ A = N{) \ e r s n,„ ,K.«, (in \hv case, whcro thoro (loos not ('xist (A' R) then = M \ ill the cas(\ wheic tlien^ tloi's not exist ( L ,S ) then A ~ N() Cousulei (A/,AO B { , A , R ) , { M , N ) ệ c As c IS contractible, applying (Pi )(P2)(/^3), we have possibilities: (a) M C No (b) M D A/i (c) (A/ iV) is uncom parable with ( M \ N i ) Consider each possibility' in deta.il by (a) we have: V t e A/, ( x ,a ) /Ỉ, Va € A \ ỈÍV (b) we have: V:r A/ \ A / i , ( r , a ) Ệ /{, Va Gand finally from r e A/, either :r G A/ n M ị c A/q, or X e A' or (.r ,a ) Ệ /?, Va G Á , respectively {Ni n N)* D In conclusion, if 7’ is an object, then 1) r e o or 2) (x, a) e /?, Va € Ấ or (c), it implies that either ( x ,a ) e /?,Va € Nguyen Due D 3) ( r ,a ) ệ R, y a E Ả Th us, o = ƠI U Ơ U Ơ , whore i , ,0'2 pairwisely disjoint and the condition of Definition 2.11 is satisfied Analogously, we have also A — A \ D A \J A -2 such th a t A \ , A , A are pairwisely disjoint ami condition (2) holds Finally, consider a pair [ x , a ) w ith X G i , a € A i;a s X G \ then /1., G /?, V/Ỉ G thus ;r G A/, VA/ D M ị O n th e other hand, as a E A i then i y , a ) G /?, Viy G o ,thus a N , y N c N ị In conclusion, (.r,a ) G R and (3) holds Now we take (Ơ , A , /? ) w ith R ~ R n o X A and D { , A R ) u (E , F) E B { \ A \ R' ) then, by (1), (2), (3) we have (Oi u E A ị U F ) e c This ronespoiuleiice is an isomorphism between B ( , A , /? ) and c The proof is complettHl Ộ P r o p o s i t i o n I f c is a contractible Sìibỉíìttice o f B { , A, R), then there exists a context siich th a t B { , A, R ) / p ~ ) (here p is H congrueiice on B { , A , R ) d e te n u in e d by C) Proof Denote by (Ơ ,i? ) the context corresponding c (proposition 2.13.) and put = \ f = A \ ^ = ( \ ') U { s } , ^ - i A \ Á ) U { t } Define between and as follows: 1) ( s , € R ^ 2) r ^ ,s,a € /?, ^ thoii(.r, a ) G 3) ( T , t ) e J ĩ ^ ^ r e O i , 4) ( s , a ) e a Ai Take p O i U ị s ị Q = 4i u { / } it is easy to deduce th a t (P ,Q ) E B ( ^ , A ^ , / ỉ ^ ) ami D { O A , R ) / p ~ D ( ^ , n ^ ) :uuh tinclaaa c of D [ A li) V corresponding to {P , Q ) of /ỉ^ ) The proposition is proved Now, we apply p ro p o s it io n s (2.13), (2.14) to s t ud y an arbitrary iinite^ foi'M xt { , A , R ) Consider its c o n ce p t lat ti ce D { , A , R ) C a s e (I) Lot B { A R) he linearly indecomposable and have contractible su blat tices Each of these sublattic’os is ornbiHlod into a maximal one Suppose th a t {C,|/ E /} is a family of all m axim al roiitiac'tible sublatticos By lemma 2.8 {C ,ị/ € /} deteiiniiK' a congruence p and a auotif^nt lattico B { , A , R ) / p , the latest has no coiitiactible sublat- tices According to (2.13) am i (2.14), there exist subcontexts — ( , , | , / Ỉ , ) (letennin- ing c , , i € L and context ( O ^ ^ R * ) deterinining B ( , A, R ) / p In this case, we* say th a t (O M ,-/?) has (leconiposition by a system of the coiit(‘xrs c , ,? E I and where c , / G I aio paiwisely disjoint and Example Consider th e lattice in Fig.3a is iiulecomposable C o n tr a c t i b le su blattices in d ata a n a ly sis C a s e ( I I ) B { , -4,/i) is not liiK’culy (l(K‘omposabl(' and lias contiaftihlc sublattia^s C ị ,C n such th a t D D C\ ^ D c „ , wlioie the next su b lattice is Iiiaxiiaal iiỉ tho previous one ^ C\ D D Ct> In this case, we have a decomposition Ermnple Coiisider D [ O A R ) in Hg.3h C ase ( I I I ) D { , A , R ) has a linear (leconiposition Suppose that C j , C V - - C n ai(* tlie linear Iiieiiibeis of B { , A, R), which are iiuearly indecomposable If, foi some ^ ^ | | > then is a contractible sublattice, w hich is a Iiiulti - dimensional part of B { A R) In rlii^ case (O, 4, R) has a convenient decom position such th a t B ( , A, R ) ! p is a linear lattice E.nifiipk Consiilcr B { , A R ) in Fift 3c c, B (0, A, R), Ci n C = 3a B (0 , A, R) = Ct u C u C 3b 3c Fig.3 lU iF E R E N C E S 1.] N gu ye n Due D a i S o m e results conceiiiiiig a G r a t z e r ’s problem, VNU J.Sct, Nat Sci, t x i , N ° (1995), 64 - 71 2.] Nguyen Due Dat On lattices L determined by Sub (L) up to isomorphism Vietnam: J Math, 24 (1996) 357 - 365 3.] D.J.N Van Eijck and Nguyen Quoc Toaii U n d e r s t a n d i n g the world: f r o m facts to concepts, fro m ('oncepts to proposttton, Centrum, voor wtskund en InformaUca rapport, cornpiittr Science Departenient of Softwaro Technology No te c s - N9301 - 1993 Nguyen Due D 4.] P Luksch aiul R.Wille Foi'iiial concept analysts of paired coĩĩrparìsoĩiò, Cla.Hstp.catron and Related nie.fhods of Data analysis In H.H Bock, editor North - Holaiul 1988, 167 - 176 [5.Ị F Vogt c Wachtor and R Wille Data analysts based Oil a conctptual file, Cldssijicatioĩi, daia analysis and knowledge orqaiiization 111 H.H.Block and P.IỈini, ('(litors Springer 1991, 131 - 140 6.Ị R W^ille Resfructurnii} ỉaỉlịce theory: an approach based on iue.rchies o f coiu epfs Ordered Sets In I Rival, (‘(litoi 1982 445 - 470 7.] R Wille SulxliifH't