General approach to triadic concept analysis

Transcription

1 General approach to tradc concept analyss Jan Konecny and Petr Osca Dept. Computer Scence Palacy Unversty, Olomouc 17. lstopadu 12, CZ Olomouc Czech Republc emal: Abstract. Tradc concept analyss (TCA) s an extenson of formal concept analyss (dyadc case) whch taes nto account mod (e.g. tme nstances, condtons, etc.) n addton to objects and attrbutes. Thus nstead of 2-dmensonal bnary tables TCA concerns wth 3-dmensonal bnary tables. In our prevous wor we generalzed TCA to wor wth grades nstead of bnary data; n the present paper we study TCA n even more general way. In order to cover up an analogy of sotone conceptformng operators (nown from dyadc case n fuzzy settng) we developed an unfyng framewor n whch both nds of concept-formng operators are partcular cases of more general operators. We descrbe the unfyng framewor, propertes of the general concept-formng operators, show ther relatonshp to those we used n our prevous wor. 1 Introducton The tradc approach to concept analyss (TCA) was ntroduced by Wlle and Lehman n [12]. TCA s an extenson of Formal Concept Analyss; t s based on a formalzaton of the tradc relaton connectng objects, attrbutes, and condtons (we recall the basc notons of TCA n Secton 2). In our prevous wor [3], we generalzed TCA for graded data (fuzzy settng). The present paper generalzes TCA even more. (Anttone) Galos connectons and concept lattces of data represented by a fuzzy relaton (graded data) were studed n a seres of papers, see e.g. [1, 14]. An alternatve approach, based on anttone Galos connectons, was studed n [10, 13]. The concept lattces based on the anttone and the sotone Galos connectons have dstnct, natural meanng. It s well nown that n the ordnary (two-valued) settng, the anttone and sotone cases are mutually reducble due to the law of double negaton and that such a reducblty fals n a fuzzy settng because the law of double negaton s not avalable n fuzzy logc. Nevertheless, a framewor whch enables a unfyng approach to both the anttone and sotone cases was recently proposed n [5, 7], see also [10]. In ths paper, nspred by ths approach, we provde a unfyng framewor that enables us to treat the sotone and anttone cases wthn TCA. We provde mathematcal foundatons of the unfyng framewor, show ts propertes, and descrbe the way t covers the two

2 General approach to tradc concept analyss 117 types of concept-formng operators. We also show that an analogy of the basc theorem holds true. The man nspraton for the presented wor, n addton to developng a general framewor whch covers dstnct partcular cases, s the recent wor n relatonal factor analyss [4, 5, 3, 6], n whch concept lattces, both the anttone and the sotone are of crucal mportance because they are optmal factors for relatonal matrx decompostons. In partcular, tradc concept lattces were shown to play the role of the space of optmal factors for factor analyss of three-way bnary data n [6]. To develop a general mathematcal framewor that can be used for a factor analyss of ordnal (graded) data s the man goal of ths paper. Proper generalzaton can smplfy defntons (as we demonstrate n Examples and ) and open door to new extenstons we ntent to use t to generalze factor analyss of three-way graded data. The paper s organzed as follows: Secton 2 recalls basc notons from (crsp) tradc concept analyss. Secton 3 descrbes the unfyng framewor and ts basc propertes. In Secton 4 we turn our attenton to tradc concept-formng operators, tradc concepts. Secton 5 brngs an analogy to basc theorem of concept (tr)-lattces. Our conclusons and future research deas are summarzed n Secton 6. 2 Prelmnares Tradc Formal Concept Analyss Ths secton ntroduces the notons needed n our paper. For further nformaton we refer to [12, 16] (tradc FCA). A tradc context s a quadruple X,Y,Z,I where X, Y, and Z are nonempty sets, and I s a ternary relaton between X, Y, and Z,.e. I X Y Z. X, Y, and Z are nterpreted as the sets of objects, attrbutes, and condtons, respectvely; I s nterpreted as the ncdence relaton ( to have-under relaton ). That s, x,y,z I s nterpreted as: object x has attrbute y under condton z. In ths case, we say that x,y,z (or x,z,y, or the result of lstng x,y,z n any other sequence) are related by I. For convenence, a tradc context s denoted by X 1,X 2,X 3,I. Let K = X 1,X 2,X 3,I be a tradc context. For {,j,} = {1,2,3} (.e.,j, {1,2,3}s.t. j ) and C X, we defne a dyadc context by K j C = X,X j,i j C x,x j I j C ff for each x C : x,x j,x are related by I. The concept-formng operators nduced by K j C are defned as follows: C (jc ) = {x j X j for each x C : x,x j I j C },

3 118 Jan Konecny and Petr Osca Operators (jc) and (jc) form a Galos connecton between X and X j [9]. A tradc concept of X 1,X 2,X 3,I s a trplet C 1,C 2,C 3 of C 1 X 1, C 2 X 2, and C 3 X 3, such that for every {,j,} = {1,2,3} we have C = C (jc ) j ; C 1, C 2, and C 3 are called the extent, ntent, and modus of C 1,C 2,C 3. The set of all tradc concepts of X 1,X 2,X 3,I s denoted by T (X 1,X 2,X 3,I) and s called the concept trlattce of X 1,X 2,X 3,I ; we refer to Secton 5 where the noton of a trlattce s defned. Fuzzy sets As a structure of truth-degrees we use a complete lattce. Gven a complete lattce L, we defne the usual notons [1, 11]: an L-set (fuzzy set, graded set) A n a unverse U s a mappng A: U L, A(u) beng nterpreted as the degree to whch u belongs to A. Let L U denote the collecton of all L-sets n U. The operatons wth L-sets are defned componentwse. For nstance, the ntersecton of L-sets A,B L U s an L-set A B n U such that (A B)(u) = A(u) B(u) for each u U, etc. We wrte A B ff A(u) B(u) for each u U. Note that 2-sets and operatons wth 2-sets can be dentfed wth ordnary sets and operatons wth ordnary sets, respectvely. Bnary L-relatons (bnary fuzzy relatons) between X and Y can be thought of as L-sets n the unverse X Y ; smlarly for ternary L-relatons. 3 Unfyng framewor In ths secton we descrbe the structure of truth degrees we use. In our prevous wor we used resduated lattces as a scale of truth degrees. Our current approach dffers n that we allow the fuzzy sets whch consttutes tradc concepts, and the nput table to have complete lattces wth common support set and dual order as ther scales of truth degrees. Moreover, we defne operatons on the structure of truth degrees that are counterparts of operatons from resduated lattces. Ths approach s nspred by [5, 7, 10]. Let L = (L, ) be a bounded complete lattce and for {1,2,3,4}, L = (L, ) be bounded lattce wth L = L and beng ether or 1. That s, each L s ether (L, ) or (L, 1 ). We denote the operatons on L by addng the subscrpt, e. g. the operatons n L 2 are denoted by 2, 2,0 2, and 1 2. We consder a ternary operaton : L 1 L 2 L 3 L 4. We assume that commutes wth suprema n all arguments. That s, for any a,a j L 1, b,b j L 2, c,c j L 3 we have ( a j,b,c) = j,b,c) 1 j J 4 j J (a (a, b j,c) = j,c) (1) 2 j J 4 j J (a,b (a,b, c j ) = j ) 3 j J 4 j J (a,b,c

4 General approach to tradc concept analyss 119 Furthermore, for,j, 1,2,3 we defne the operatons : L j L L 4 as (a j,a,a 4 ) = {a (a,a j,a ) a 4 } (2) For convenence we denote (a,b,c) also by {b,a,c} or {c,b,a} etc., and (a,a j,a 4 ) also by {a j,a,a 4 } or {a,a j,a 4 } Example 1. Complete resduated lattce [11] s an algebra L = L,,,,,0,1 such that L,,,0,1 s a complete lattce wth 0 and 1 beng the least and greatest element of L, respectvely; L,,1 s a commutatve monod (.e. s commutatve, assocatve, and a 1 = a for each a L); and satsfy so-called adjontness property: a b c ff a b c for each a,b,c L. Let L = L,,,,,0,1 be a complete resduated lattce and be ts order. (1) Let L = L j = (L, ) for each,j {1,2,3,4} and let (a 1,a 2,a 3 ) = a 1 a 2 a 3. Then operatons are defned as follows: 1 (a 2,a 3,a 4 ) = (a 2 a 3 ) a 4 (3) 2 (a 3,a 1,a 4 ) = (a 3 a 1 ) a 4 (4) 3 (a 1,a 2,a 4 ) = (a 1 a 2 ) a 4 (5) (2) Let L 1 = L 2 = L, and L 3 = L 4 = L, 1 and let (a 1,a 2,a 3 ) = (a 1 a 2 ) a 3. Then operatons are defned as follows 1 (a 2,a 3,a 4 ) = (a 4 a 2 ) a 3 (6) 2 (a 3,a 1,a 4 ) = (a 4 a 1 ) a 3 (7) 3 (a 1,a 2,a 4 ) = a 1 a 2 a 4 (8) We show usablty of both sets of operators n Example 2. The followng lemma descrbes basc propertes of the prevously defned operatons that we wll need n rest of the paper. Lemma 1. x s monotone n all arguments. (a) (b) are monotone n frst two arguments and anttone n thrd argument. (c) (a 1,a 2, 3 (a 1,a 2,a 4 )) a 4, analogous formulas hold for 1 and 2. (d) 3 (a 1,a 2, (a 1,a 2,a 3 )) a 3, analogous formulas hold for 1 and 2. Proof. (a) follows drectly from (1) (b) follows drectly from (2) (c) (a 1,a 2, 3 (a 1,a 2,a 4 )) = = (a 1,a 2, 3 {a 3 (a 1,a 2,a 3 ) a 4 }) = = { (a 1,a 2,a 3 ) (a 1,a 2,a 3 ) a 4 } a 4 4

5 120 Jan Konecny and Petr Osca (d) 3 (a 1,a 2, (a 1,a 2,a 3 )) = = {x 3 (a 1,a 2,x 3 ) (a 1,a 2,a 3 )} a Tradc context, concept-formng operators, and concepts In ths secton we develop the basc notons of the general approach to tradc concept analyss. We defne the notons of L-context, concept-formng operators and tradc concepts n our settng and nvestgate ther propertes. Tradc L-context s a quadruple X,Y,Z,I where X, Y, Z are non-empty sets nterpreted as sets of objects, attrbutes, and condtons, respectvely. I s a ternary L-relaton between X, Y and Z,.e.: I : X Y Z L 4. For every x X, y Y, and z Z, the degree I(x,y,z) n whch are x,y, and z related s nterpreted as the degree to whch object x has attrbute y under condton z. For convenence, we denote I(x,y,z) also by I{x,y,z} or I{x,z,y} or I{z,x,y}, and the tradc L-context by X 1,X 2,X 3,I. L-context K = X 1,X 2,X 3,I nduces three concept-formng operators. For {,j,} = {1,2,3} and the sets A L X and A L X, the concept-formng operator s a map: L L L 4 L j whch assgns to A and A a fuzzy set A j L Xj defned by A j (x j ) = j {A (x ),A (x ),I{x,x j,x }}. (9) j x X In ths case, the concept-formng operator s denoted by (ja ),.e. fuzzy set A j s denoted by A j = A (ja ). Example 2. (1) Let L and be as n Example 1(1). Then the concept-formng operators are as follows A (ja ) (x j ) = (A (x ) A (x )) I(x 1,x 2,x 3 ) (10) x X for {,j,} {1,2,3}. Note that these operators are fuzzy generalzatons of those descrbed n Secton 2. These concept-formng operators also appear n [8]. (2) Let L and be as n Example 1(2). Then the concept-formng operators are defned as follows: A (12A3) 1 (x 2 ) = x 1 X 1 x 3 X 3 (I(x 1,x 2,x 3 ) A 1 (x 1 )) A 3 (x 3 ) (11) A (23A1) 2 (x 3 ) = x 1 X 1 x 2 X 2 (I(x 1,x 2,x 3 ) A 2 (x 2 )) A 1 (x 1 ) (12) A (31A2) 3 (x 1 ) = x 2 X 2 x 3 X 3 (I(x 1,x 2,x 3 ) A 1 (x 1 ) A 3 (x 3 )) (13)

6 General approach to tradc concept analyss 121 Note that operators (12) (13) are selected as a tradc counterpart to (dyadc) sotone galos connectons [10]. Formulas (12) (13) are rather complcated n comparsson wth the general defnton (9). A tradc fuzzy concept of X 1,X 2,X 3,I s a trplet C 1,C 2,C 3 consstng of fuzzy sets C 1 L X1 1, C 2 L X2 2, and C 3 L X3 3, such that for every {,j,} = {1,2,3} we have C = C (jc ) j, C j = C (jc), and C = C (Cj). The C 1, C 2, and C 3 are called the extent, ntent, and modus of C 1,C 2,C 3. The set of all tradc concepts of K = X 1,X 2,X 3,I s denoted by T (X 1,X 2,X 3,I) and s called the concept trlattce of K. We vew the tradc concepts as trplets of fuzzy sets of objects, attrbutes, and mod. That s, a concept apples to objects to degrees; smlarly for attrbutes and condtons. In our settng, the scales of truth degrees n whch objects belong to extent, attrbutes belong to ntent, and condtons belongs to modus are complete lattces whch conssts of common support set, but they may be ordered dually. The followng lemma descrbes basc propertes of concept-formng operators. Lemma 2. (a) A (jc ) = C (ja) (b) f C D and A B then B (jd ) A (jc ) (c) A (A (ja ) ) (ja ) Proof. (a) (b) (c) A (jc ) (x j ) = j (A (x ),C (x ),I{x,x,x j }) = C (ja) (x j ) j x X B (jd ) (x j ) = j (B (x ),D (x ),I{x,x,x j }) j x X (A (ja ) ) (ja ) (x ) = j (A (x ),C (x ),I{x,x,x j }) = A (jc ) j x X = ( j (A (x ),A (x ),I{x,x,x j }),A (x ),I{x,x j,x }) xj Xj j x x X X x X ( j (A (x ),A (x ),I{x,x,x j }),A (x ),I{x,x j,x }) = xj Xj = {a ( j (A (x ),A (x ),I{x,x,x j }),x,a ) I{x,x,x j }} xj Xj

7 122 Jan Konecny and Petr Osca Lemma 1(c) yelds that one of the possble values of a s A (x ). Therefore, the prevous formula s greater than x j X j A (x ) = A (x ) whch concludes the proof. Theorem 1. Let {,j,} = {1,2,3}. Then for all tradc fuzzy concepts A 1,A 2,A 3 and B 1,B 2,B 3 from T (K), f A 1,A 2,A 3 B 1,B 2,B 3 and A 1,A 2,A 3 j B 1,B 2,B 3 then B 1,B 2,B 3 A 1,A 2,A 3. Proof. We have A = A (Aj) Lemma 2 yelds B A. and B = B (Bj). Snce A B and A j B j, The followng theorem descrbes a way how to compute a tradc concept. Startng wth two fuzzy sets C L X and C L X we obtan a tradc concept A 1,A 2,A 3 by three projectons usng the concept-formng operators. Frstly we project C and C onto A j, then we project A j and C onto A, and fnally we project A and A j onto A Theorem 2. For C L X,C L X wth {,j,} = {1,2,3}, let A j = C (jc ), A = A (jc ) j, and A = A (Aj). Then A 1,A 2,A 3 s a tradc fuzzy concept b (C,C ). Moreover, A 1,A 2,A 3 has the smallest -th component among all tradc fuzzy concepts B 1,B 2,B 3 wth the greatest j-th component satsfyng C B and X = C B. In partcular, b (A,A ) = A 1,A 2,A 3 for each tradc fuzzy concept A 1,A 2,A 3. Proof. By lemma 2(c) we have C A and snce A = A (Aj) = (A (jc ) j ) (Aj) = (C (Aj) ) (Aj) we have also C A. Frst, we prove that A 1,A 2,A 3 s a tradc fuzzy concept. A = A (Aj) s satsfed by defnton. Consder A j. We have A j = C (jc ) A (ja ) (Lemma 2(b)) and A j (A (ja) j ) (ja) = A (ja) = A (ja ). Therefore, A j = A ja. The proof for A s smlar. Let B 1,B 2,B 3 be a tradc fuzzy concept wth X B and X B. Then B j = B (jb ) X (jx ), so the maxmal j-th component s A j. Let B j = A j. Then A = A (jx ) j B (jb ) j = B and thus A = A (Aj) B (Bj) = B. The last asserton s easly observable from the defnton of tradc fuzzy concept. In the rest of the paper we need the followng notaton. For fuzzy sets A 1 L X1 1, A 2 L X2 2, and A 3 L X3 3 we denote by A 1 A 2 A 3 the ternary L 4 -relaton between X 1, X 2, and X 3 defned by (A 1 A 2 A 3 )(x 1,x 2,x 3 ) = (A 1 (x 1 ),A 2 (x 2 ),A 3 (x 3 )). The followng lemma descrbes a geometrc vew on tradc fuzzy concepts,.e. that tradc fuzzy concepts can be vewed as maxmal clusters contaned n the nput data. Lemma 3. (a) If A 1,A 2,A 3 T (K) then A 1 A 2 A 3 I.

8 General approach to tradc concept analyss 123 (b) If A 1 A 2 A 3 I then there s B 1,B 2,B 3 T (K) such that A B for = 1,2,3. (c) Each A 1,A 2,A 3 T (K) s maxmal w.r.t. to set ncluson,.e. there s no B 1,B 2,B 3 T (K) other than A 1,A 2,A 3 for whch A B. Proof. (a) (A (x ),A j (x j ), x X (A (x ),A j (x j ),I{x,x j,x }) (A (x ),A j (x j ), (A (x ),A j (x j ),I{x,x j,x })) I{x,x j,x } (b) Let {,j,} = {1,2,3} and b (A,A ) = B 1,B 2,B 3. Due Theorem 2 we have A B and A B. Moreover, B j (x j ) = A (ja ) (x j ) = = j (A (x ),A (x ),I{x,x,x j }) j x X j x X = A j (x j ) j (A (x ),A (x ), (A (x ),A (x ),A j (x j ))) = (c) Let A 1,A 2,A 3 and B 1,B 2,B 3 be tradc concepts wth A B and {1,2,3} be an ndex such that A j B j. From Theorem 1 follows that there s an ndex j {1,2,3} such that A = B. Then havng A j = A (ja ) B j = B (jb ) Lemma 2(c) yelds B j A j whch s a contradcton. Theorem 3 (crsp representaton). Let K = X 1,X 2,X 3,I be a fuzzy tradc context and K crsp = X 1 L 1,X 2 L 2,X 3 L 3,I crsp wth I crsp defned by ((x 1,a),(x 2,b),(x 3,c)) I crsp ff (a,b,c) 4 I(x 1,x 2,x 3 ) be a tradc context. Then T (K) s somorphc to T (K crsp ). Proof. Defne maps... : L X X L and... : X L L X for {1, 2, 3} as follows: A = {(x,a ) a A (x )} (14) A = {a (x,a ) A } (15) In what follows we sp subscrpts and wrte just A and A nstead of A and A. Let ϕ be a mappng ϕ : T (K) T (K crsp ) defned by ϕ( A 1,A 2,A 3 ) = A 1, A 2, A 3. We show, that ϕ( A 1,A 2,A 3 ) T (K crsp ). We have (x,b) ( A j (j A ) ff for each ((x j,a),(x,c)) A j A t holds that ((x,b),(x j,a),(x,c))

9 124 Jan Konecny and Petr Osca I crsp ff for each x j X j,x X, and for each a j A j (x j ),b A (x ) t holds (a,b,c) 4 I{x,x j,x } ff for each x j X j,x X we have (A j (x j ),A (x ),b) 4 I{x,x j,x } ff b A (x ), therefore ( A j (j A ) A ) = A. Let ψ be a mappng ψ : T (K crsp ) T (K) defned by ψ( A 1,A 2,A 3 ) = A 1, A 2, A 3. We show, that ψ( A 1,A 2,A 3 ) T (K). We have ( A j (j A ) (x ) = b ff b s the maxmal degree wth the property that for each x j X j,x X t holds ( A j (x j ), A (x j ),b) 4 I(x,x j,x ) ff b s the maxmal degree wth the property that for each a b and each x j X j,x X we have ( A j (x j ), A (x j ),a) 4 I(x,x j,x ) ff b s the maxmal degree wth the property that for each a b and each ((x j,c),(x,d)) A j A we have that ((x,a),(x j,c),(x,d)) I crsp ff b s the maxmal degree wth the property that for each a b we have (x,a) A (ja ) j = A. Therefore A j (j A ) = A. Snce A = A for each fuzzy set A, the mappngs ϕ and ψ are mutually nverse and ϕ s a bjecton. Moreover, A B ff A B for all fuzzy sets A and B and thus ϕ preserves 1, 2, 3. 5 Basc theorem In ths secton, we defne mportant structural relatons on the set of tradc concepts. These relatons are based on the subsethood relatons on the sets of objects, attrbutes, and mod, and are fundamental for an understandng of the structure of the set of all tradc concepts. In the fnal part of ths secton, we prove a theorem whch s a generalzaton of the basc theorem of tradc concept analyss [16]. Consder the followng relatons A 1,A 2,A 3 B 1,B 2,B 3 ff A B, A 1,A 2,A 3 B 1,B 2,B 3 ff A = B. It s easy to chec that and are a quasorder and an equvalence on T (K). Denote by T (K)/ the correspondng factor set wth equvalence classes denoted by [ A 1,A 2,A 3 ]. Lettng [ A 1,A 2,A 3 ] [ B 1,B 2,B 3 ] ff A 1,A 2,A 3 B 1,B 2,B 3, s an order on T (K)/. Let V be a non-empty set, and for {1,2,3} let be quasorder relatons on V. Then we call (V, 1, 2, 3 ) a trordered set f and only f t holds that v w and v j w mples w v for {,j,} = {1,2,3} and each v,w V and j ( = ) s an dentty relaton. Clearly, = s an equvalence, and j s an dentty relaton on V. Moreover, turns nto an orderng on V/ and so (V/, ) s an ordered set. An element v V s an -bound of (V,V ), V,V V, f x v for all x V and x v for all x V. An -bound v s called an -lmt of (V,V ) f

10 General approach to tradc concept analyss 125 u j v for all -bounds of (V 1,V 2 ) u. In an trordered set (V, 1, 2, 3 ) there s at most one -lmt of (V 1,V 2 ) v wth a property u v for all -lmts of (V 1,V 2 ) u. Then we call v an -jon of (V,V ) and denote t (V,V ). The trordered set (V, 1, 2, 3 ) n whch the -jon exsts for all and all pars of subsets of V s a complete trlattce. For a complete trlattce V = (V, 1, 2, 3 ), an order flter F on ordered set V/ s defned as a subset F of V wth the property: x F and x y mples y F for all x,y V. We denote the set of all order flters on V/ by F (V). A prncpal flter generated by x V s the flter [X) = {y V x y}. We call a subset X F (V) of flters -dense wth respect to V f each prncpal flter of (V,/ ) can be obtaned as an ntersecton of some order flters from X. It s easy to see that T (K) s a trordered set. Let κ : X L T (K) be a mappng defned by κ (x,b) = { A 1,A 2,A 3 T (K) A (x ) b} for {1,2,3}, x X and b L. Snce the prncpal flter generated by A 1,A 2,A 3 s [ A 1,A 2,A 3 ) = x X κ (x,a (x )), the set κ (X L ) s -dense. Moreover, κ happens to satsfy κ (x,a) κ (x,b) ff b a. Theorem 4 (basc theorem). Let K = (X 1,X 2,X 3,I) be a fuzzy tradc context. Then T (K) s a complete trlattce of K for whch the -jons are defned as follows: ( (X, X j ) = b {A A 1,A 2,A 3 X }, ) {A A 1,A 2,A 3 X }. A complete trlattce V = (V, 1, 2, 3 ) s somorphc to T (K) f and only f there are mappngs κ : X L F (V), = 1,2,3, such that (a) κ (X L ) s -dense wth respect to V, (b) κ (x,a) κ (x,b) ff b a, (c) A 1 A 2, A 3 I 3 =1 x X κ (x,a (x )) for all A L X. Proof. The frst asserton follows from Theorem 2. From Theorem 3 we now that T (K) s somorphc to T (K crsp ). To prove our asserton t suffces to show that condtons (a),(b), and (c) (for T (K)) are equvalent wth the condtons from Wlle s orgnal basc theorem (for T (K crsp )). Consder the map κ w : (X L ) F (V) defned by κ w ((x,a)) = κ (x,a). Obvously, κ w s -dense ff κ s -dense. Furthermore, we have A 1 A 2 A 3 I A 1 A 2 A 3 I crsp, and snce f a b then κ w ((x,b)) κ w ((x,a)), we obtan Ths concludes the proof. 3 =1 (x,a) A κ w ((x,a)) 3 =1 x X κ w (x, {c (x,c) A }) 3 =1 x X κ (x,a (x ))

11 126 Jan Konecny and Petr Osca 6 Concluson We presented how foundatons of tradc concept analyss can be developed n a very general way. We showed that the prevously studed cases of fuzzy TCA, namely the TCA wth sotone and TCA wth anttone concept-formng operators, are just partcular cases of a more general approach. We provded defntons of basc notons, descrbed propertes of concept-formng operators and tradc concepts, and proved the analogy of basc theorem of TCA usng crsp representaton of tradc concepts. Our future research topcs on general approach to TCA nclude: Investgaton of attrbute mplcatons n the unfyng framewor we have developed. At the current moment we study attrbute mplcatons n a unfyng framewor for dyadc case. Generalzaton of the unfyng framewor assumng supports of the lattces L to be dfferent. References 1. Belohlave R.: Fuzzy Relatonal Systems: Foundatons and Prncples. Kluwer, Academc/Plenum Publshers, New Yor, Belohlave R.: Concept lattces and order n fuzzy logc. Annals of Pure and Appled Logc 128(1 3)(2004), Belohlave R., Vychodl V.: Factor analyss of ncdence data va novel decomposton of matrces. Lecture Notes n Artfcal Intellgence 5548(2009), Belohlave R.: Optmal trangular decompostons of matrces wth entres from resduated lattces. Int. J. of Approxmate Reasonng 50(8)(2009), Belohlave R.: Optmal decompostons of matrces wth entres from resduated lattces. Condtonally accepted to J. Logc and Computaton. 6. Belohlave R., Vychodl V.: Optmal factorzaton of three-way bnary data. Proc. of The 2010 IEEE Internatonal Conference on Granular Computng, 2010, San Jose. 7. Belohlave R.: Sup-t-norm and nf-resduum are one type of relatonal product: Unfyng framewor and consequences. Preprnt to be submtted. 8. Belohlave R., Osca P.: Tradc concept analyss of data wth fuzzy attrbutes. Proc. of The 2010 IEEE Internatonal Conference on Granular Computng, 2010, San Jose. 9. Ganter B., Wlle R.: Formal Concept Analyss. Mathematcal Foundatons. Sprnger, Berln, Georgescu G., Popescu A.: Non-dual fuzzy connectons. Archve for Mathematcal Logc 43 (2004). 11. Háje P.: Metamathematcs of Fuzzy Logc. Kluwer, Dordrecht, Lehmann F., Wlle R.: A tradc approach to formal concept analyss. Lecture Notes n Computer Scence 954(1995), Popescu A.: A general approach to fuzzy concepts. Math. Log. Quart. 50(2004), Pollandt S.: Fuzzy Begrffe. Sprnger-Verlag, Berln/Hedelberg, Ward M., Dlworth R. P.: Resduated lattces. Trans. Amer. Math. Soc. 45(1939), Wlle R.: The basc theorem of tradc concept analyss. Order 12(1995), Zadeh L. A.: Fuzzy sets. Inf. Control 8(1965),