One of the main problems in FCA (especially in fuzzy setting) is to reduce a concept lattice to appropriate size to make it graspable and understandable. We generalize known results on the correspondence of block relations of formal contexts and complete tolerances on concept lattices to fuzzy setting. Using an illustrative example we show that the results can be used for reduction of fuzzy concept lattices.
The problem of the size of concept lattices is recognized as one of the most important problems of Formal Concept Analysis (FCA). The main aim of this paper is to describe approximation of formal concepts by fuzzy tolerances in fuzzy setting.
In [
In [
The organization of the paper is as follows. In Section 2 we recall the fundamental notions and results on the basic fuzzy structures used in the paper. Section 3 is devoted to study of fuzzy tolerance relations: we define complete tolerance relations on completely lattice ordered fuzzy sets and show fundamental properties of the corresponding factor sets. In Section 4 we describe fuzzy block relations and prove the main result of the paper: the correspondence between fuzzy block relations and factor structures of concept lattices. Section 5 contains an illustrative example.
Due to space limitations, we abbreviate, resp. omit, some proofs. 2
We recall basic notions and facts of residuated lattices, fuzzy sets and fuzzy relations, fuzzy Galois connections and fuzzy concept lattices. 2.1
We use complete residuated lattices as basic structures of truth degrees. A complete
residuated lattice [
and ! satisfy adjointness, i.e., a b c iff a b ! c. 0 and 1 denote the least and greatest elements. The partial order of L is denoted by . Throughout the paper, L denotes an arbitrary complete residuated lattice.
Elements a of L are called truth degrees. and ! (truth functions of) “fuzzy conjunction” and “fuzzy implication”.
Common examples of complete residuated lattices include those defined on a unit interval, (i.e., L = [0; 1]) or on a finite chain in the unit interval, e.g. L = f0; 1n ; : : : ; n n 1 ; 1g, ^ and _ being minimum and maximum, being a left-continuous t-norm with the corresponding !.
The three most important pairs of adjoint operations on the unit interval are Łukasiewicz: aa! bb == mmianx((1a +ab+ b1;;10))
a b = min(a; b) Go¨del: 1 a b a ! b = b otherwise a b = a b Goguen (product): 1 a b
a ! b = ba otherwise
An L-set (or fuzzy set) A in a universe set X is a mapping assigning to each x 2 X some truth degree A(x) 2 L where L is the support of a complete residuated lattice. The set of all L-sets in a universe X is denoted LX .
The operations with L-sets are defined componentwise. For instance, the intersection of L-sets A; B 2 LX is an L-set A \ B in X such that (A \ B)(x) = A(x) ^ B(x) for each x 2 X , etc. An L-set A 2 LX is also denoted fA(x)=x j x 2 X g. If for all y 2 X distinct from x1; x2; : : : ; xn we have A(y) = 0, we also write fA(x1)=x1; A(x2)=x1; : : : ; A(xn)=xng.
Binary L-relations (binary fuzzy relations) between X and Y can be thought of as L-sets in the universe X Y . That is, a binary L-relation I 2 LX Y between a set X and a set Y is a mapping assigning to each x 2 X and each y 2 Y a truth degree I(x; y) 2 L (a degree to which x and y are related by I). An L-set A 2 LX is called crisp if A(x) 2 f0; 1g for each x 2 X . Crisp L-sets can be identified with ordinary sets. For a crisp A, we also write x 2 A for A(x) = 1 and x 62 A for A(x) = 0. An L-set A 2 LX is called empty (denoted by 0/ ) if A(x) = 0 for each x 2 X . For a 2 L and A 2 LX , a A 2 LX and a ! A 2 LX are defined by (a
A)(x) = a
A(x) and (a ! A)(x) = a ! A(x):
For an L-set A 2 LX and a 2 L, the a-cut of A is a crisp subset aA X such that x 2 aA iff a A(x). This definition applies also to binary L-relations, whose a-cuts are interpreted as classical (crisp) binary relations.
For universe X we define L-relation graded subsethood LX
LX ! L by: S(A; B) = ^ A(x) ! B(x)
x2X Graded subsethood generalizes the classical subsethood relation ; indeed, in the crisp case (i.e. L = f0; 1g) (1) becomes S(A; B) = 1 iff 8x 2 X : x 2 A implies y 2 B. Note that unlike , S is a binary L-relation on LX . Described verbally, S(A; B) represents a degree to which A is a subset of B. In particular, we write A B iff S(A; B) = 1. As a consequence, we have A B iff A(x) B(x) for each x 2 X .
Further we set
A
X B = S(A; B) ^ S(B; A): The value A X B is interpreted as the degree to which the sets A and B are similar.
A binary L-relation R on a set X is called reflexive if R(x; x) = 1 for any x 2 X , symmetric if R(x; y) = R(y; x) for any x; y 2 X , and transitive if R(x; y) R(y; z) R(x; z) for any x; y; z 2 X . R is called an L-tolerance, if it is reflexive and symmetric, L-equivalence if it is reflexive, symmetric and transitive. If R is an L-equivalence such that for any x; y 2 X from R(x; y) = 1 it follows x = y, then R is called an L-equality on X . Lequalities are often denoted by . The similarity X of L-sets (2) is an L-equality on LX .
Let be an L-equivalence on X , A 2 LX an L-set. We say that A is compatible with (or extensional with respect to ) if the following condition holds for any x; x0 2 X : A(x) (x x0)
A(x0): For each A 2 LX there exists a minimal (with respect to inclusion of L-sets) L-set C (A) 2 LX , compatible with , and such that A C (A). It holds
C (A)(x) = _ A(x0) (x0
x): x02X We say that a binary L-relation R on X is compatible with if for each x; x0; y; y0 2 X , R(x; y) (x x0) (y
For a mapping f : X ! Y , where Y is endowed with an L-equality , and L-set
f (A)(y) = _ A(x) ( f (x) x2X y): f (A) is compatible with and f (A) = C (B), where B is is an L-set in Y given by B(y) = Wx2 f 1(fyg) A(x).
In the following we use well-known properties of residuated lattices and fuzzy
structures which can be found e.g. in [
In this section, we recall basic definitions and results of the theory of L-ordered sets.
Basic references are [
An L-order on a set U with an L-equality relation is a binary L-relation on U which is compatible with , reflexive, transitive and satisfies (u v) ^ (v u) u v for any u; v 2 U (antisymmetry). The tuple U = hhU; i; i is called an L-ordered set. An immediate consequence of the definition is that for any u; v 2 U it holds u v = (u
Shifts and multiples satisfy the following adjointness condition: If a ! v and a u both exist then (u (a ! v)) = ((a u) v).
The following theorem has been proved in [
Theorem 1. V is a completely lattice L-ordered set iff V is shift complete and multiple complete and hV; 1 i is a complete lattice.
The notions of shift and multiple coincide with the notions of cotensor and tensor
[
For any u; v 2 U and a 2 L it holds v = a ! u iff for each w 2 U , (5) (6) (7) (8)
We introduce the notion of isotone L-Galois connections. For details, see [
An isotone L-Galois connection between L-ordered sets U and V is a pair h f ; gi, where f : U ! V , g : V ! U are mappings such that for each u 2 U , v 2 V it holds ( f (u) v) = (u g(v)): A pair hu; vi, where u 2 U and v 2 V , is called a fixpoint of h f ; gi if f (u) = v and g(v) = u. More generally, the degree to which hu; vi is a fixpoint of h f ; gi is defined by Fixh f ;gi(hu; vi) = ( f (u) v) ^ (g(v) u): Fixh f ;gi is an L-set in U
V and is called the L-set of fixpoints of h f ; gi.
Theorem 2 (basic properties of isotone L-Galois connections). Let h f ; gi be an isotone L-Galois connection between L-ordered sets U and V. Then (a) u g( f (u)) for each u 2 U , f (g(v)) v for each v 2 V .
(b) f and g are increasing: (u1 u2) ( f (u1) f (u2)) and (v1 v2) (g(v1) g(v2)).
(c) f (g( f (u))) = f (u), g( f (g(v))) = g(v).
(d) If U 0 2 LU then f (infU 0) inf f (U 0) and f (supU 0) sup f (U 0). If V 0 2 LV then g(infV 0) inf g(V 0) and g(supV 0) sup g(V 0).
(e) If U and V are completely lattice L-ordered sets and K 2 LU V , then S(K; Fixh f ;gi) S(K; Fixh f ;gi)
Fixh f ;gi(hinf prU (K); f (g(inf prV (K)))i); Fixh f ;gi(hg( f (sup prU (K))); sup prV (K)i); (9) (10) where prU : U
V ! U and prV : U
V ! V are Cartesian projections.
Proof. Omitted. Some of the properties are proved in [
For an L-ordered set U, an isotone L-Galois connection h f ; gi between U and U is called an inflationary Galois connection on U if f is deflationary and g inflationary: f (u) u and g(u) u for each u 2 U . Notice that if one of the conditions (11) holds true, then the second one follows from (7). 2.4
We use three relational composition operators, , /, and .. The composition operators are defined by (A B)(x; y) = Wz2Z A(x; z) (A / B)(x; y) = Vz2Z A(x; z) ! B(z; y); (A . B)(x; y) = Vz2Z B(z; y) ! A(x; z):
B(z; y);
Note that these operators were extensively studied by Bandler and Kohout, see e.g.
[
A formal L-context (or just context) is a triple hX ;Y; Ii where X and Y are sets whose elements are called objects and attributes, respectively and I is an L-relation between X and Y . Consider the following pairs of operators between LX and LY induced by an L-relation I 2 LX Y : hA1; B1i for A 2 LX , B 2 LY . We call (15) antitone concept-forming operators and (16), (17) isotone concept-forming operators (these are less common if FCA). Furthemore, denote the set of fixpoints of h"; #i by B(X ";Y #; I). We have
B(X ";Y #; I) = fhA; Bi j A" = B; B# = Ag;
B(X ";Y #; I) is a completely lattice L-ordered set with the L-equality
defined by
and L-order
and is called the L-concept lattice associated to I. Its elements are called formal
Lconcepts. L-concept lattices are the fundamental structures of formal concept analysis
in fuzzy setting [
Y B = B#"g:
Int(X ";Y #; I) = fB 2 L j
The above-defined operators and their sets of fixpoints have been extensively studied,
see e.g. [
Example 1. Let L be the 6-element Łukasiewicz chain (i.e., L = f0; 0:2; 0:4; 0:6; 0:8; 1g) and hX ;Y; Ii be a formal L-context, where X = fBV; LH; MD; TSSg, Y = fc1; c2; c3; c4; c5g, and I is depicted in Fig. 1. Elements of X are four selected movies by director David Lynch, elements of Y are five film critics and values of I are ratings the critics assigned to the movies, taken from www.metacritic.com and rescaled to the six-element scale. The context is our central example in this paper and we describe it deeper and work with it in Section 5. For now, we use it just to give an example of an L-concept lattice.
Note that our data are suitable for interpreting by means of fuzzy logic. A movie rating (usually given by number of “stars” or by a percentage) can be interpreted as an answer to the question: “Do you like this movie?”, given in degrees. In most cases, it is not possible to answer the above question with simple “Yes” or “No”, and, in the same time, there are no doubts about the answer (especially, if given by a film critic).
BV LH MD TSS
The L-concept lattice B(X ";Y #; I) is depicted in Fig. 2. When displaying L-concept
lattices, we use labeled Hasse diagrams to include all the information on extents and
intents. For any x 2 X , y 2 Y and formal L-concept hA; Bi we have A(x) a and B(y) b
if and only if there is a formal concept hA1; B1i hA; Bi, labeled by a=x and a formal
concept hA2; B2i hA; Bi, labeled by b=y. We use labels x resp. y instead of 1=x resp.
1=y and omit redundant labels (i.e., if a concept has both the labels a=x and b=x then
we keep only that with the greater degree; dually for attributes). The whole structure
of L-ordered set on B(X ";Y #; I) can be determined from the labeled diagram using the
results from [
For an L-tolerance on a set X , an L-set B 2 LX is called a block of the L-tolerance if for each x1; x2 2 X it holds B(x1) B(x2) (x1 x2). A block B is called maximal if for each block B0 from B B0 it follows B = B0. The set of all maximal blocks of always exists by Zorn’s lemma, is called the factor set of X by and denoted by X = .
Further we set for each x 2 X , JxK (y) = x y, obtaining an L-set JxK called the class of determined by x.
An L-tolerance on a completely lattice L-ordered set U = hhU; i; i is called complete if it is compatible with and for each R it holds sup(pr1(R)) sup(pr2(R)) = inf(pr1(R)) inf(pr2(R)) = 1; (20) where pr1 (resp. pr2) is the projection U U ! U to the first (resp. second) component.
Note that in the classical case the condition (20) becomes the well-known condition
of completeness of [
For the rest of this section we suppose is a complete L-tolerance on a completely lattice L-ordered set U = hhU; i; i.
v = 1 then (a u)
(a Proof. Follows directly from the definition. v) = 1 and (a ! u) (a ! v) = 1.
Lemma 2. If is a complete L-tolerance on a complete lattice L-ordered set U = hhU; i; i then so is a ! , for each a 2 L.
Proof. Since is compatible with then for u; u0; v; v0 2 U we have (u u0) (a ! (u0 v0)) (v0 v) a ! ((u u0) (u0 v0) (v0 v)) a ! (u v); proving a ! is compatible with .
Let R a ! . We have for each u; v 2 U , (u R v) a ! (u v), which is equivalent to a (u R v) (u v). Thus, a R and we can use (20). We have sup(pr1(a R)) = sup(a pr1(R)) = a sup(pr1(R)), inf(pr1(a R)) = inf(a pr1(R)) = a ! inf(pr1(R)) and similarly for pr2. Now the result follows from Lemma 1.
complete tolerance on the complete lattice hU; 1 i. is a Proof. For a = 1 the assertion follows easily from the definition of complete L-tolerances. The rest follows from Lemma 2.
We set for any u 2 U , u = supJuK ; u = infJuK : (21) Theorem 4. The pair h ; i is an inflationary isotone L-Galois connection on U. Proof. We show that for u; v 2 U it holds (v u) (v u ), the proof of the opposite inequality is similar. Denote (v u) = a. Since U fa=v g(u) = 1 (by the definition of upper cone), then we have a v u. Now, (a v) (a v ) = 1 (Lemma 1) and u (u _ (a v)) = 1. Thus, u _ (a v) u which means that (a v) u or v u a. The fact that h ; i is inflationary is trivial.
Lemma 3. Let u; v 2 U be such that v w1) and if v w2, then (w2 u) (w2 u, u v).
Proof. By completeness of , (u _ w1) (v _ w1) = 1. Thus, v = 1. If w1 u, then (v (u Similarly for w2.
Proof. The inclusion JuK [u ; u ] follows from basic properties of cones (for exampWlee, JpurKove thLeoUppJousKite =incLlufsiuong. )F.or w 2 U denote u w = a and w u = b. We have [u ; u ](w) = a ^ b. By Lemma 3, (u (w ^ u)) ((w ^ u) u) and ((w _ u) u ) ((w _ u) u). But u (w ^ u) = a and (w _ u) u = b which means that [u ; u ](w) = a ^ b ((w ^ u) u) ^ ((w _ u) u). Using Theorem 3 we easily obtain that the right-hand side is less than or equal to w u.
are exactly L-intervals [v; u] where hu; vi are fixTheorem 6. Maximal blocks of points of h ; i.
Proof. Let w1; w2 2 U . We have [v; u](w1) [v; u](w2) = ((v = ((v ((v (w2 w1) (w2 w1) (w2 w1) ^ (w1 u)) ^ ((w1 v)) ^ ((w1 w1) ^ (w1 u) (v
u) (u w2 ) = Jw2K (w1) = (w1
w2); w2))
w2 )) u)) ((v w2) ^ (w2 u)) showing [v; u] is a block.
Conversely, if W 2 LU is a block then so is the interval [infW; supW ] which contains W as a subset. Among all intervals [v; u], the maximal blocks are those satisfying v = u and u = v.
We define two binary L-relations and on U =
by (the equalities in the parentheses follow directly from (4), Theorem 4, and Theorem 2 (b)) Theorem 7. The tuple hhU = ; i; i is a completely lattice L-ordered set. Proof. Sketch: we use Theorem 1, (5), (6).
Let h f ; gi be an inflationary Galois connection on U. Set for each u; v 2 U u h f ;gi v = ( f (u) Theorem 8. h f ;gi is an L-complete tolerance, satisfying u h f ;gi = f (u), u h f ;gi = g(u), for each u 2 U . The assignment h f ; gi 7! h f ;gi is a bijection between the set of all inflationary isotone L-Galois connections U and the set of all L-complete tolerances on U.
Proof. Sketch: We use basic properties of isotone L-Galois connections (Theorem 2) and the definition of an L-complete tolerance on U. 4
This section introduces block L-relations on L-formal contexts, their properties and the relationship to complete L-tolerances on L-concept lattices. The main results are contained in Theorems 11 and 12, where we show that block L-relations of any formal L-context are in one-to-one correspondence with complete L-tolerances on the associated L-concept lattice and that the L-concept lattice of each of the block relations is isomorphic to the original L-concept lattice, factorized by the corresponding complete L-tolerance.
We define block relations as follows: Block L-relation of I 2 LX Y is an L-relation J I such that Ext(X ";Y #; J) Ext(X ";Y #; I) and Int(X ";Y #; J) Int(X ";Y #; I).
In crisp setting, [
x 2 X and fyg#I 2 Ext(X ";Y #; I) for each y 2 Y .
The following theorem provide characterization of block L-relations.
(a) J is a block relation of I.
(b) J = I . Si with Si 2 LY Y and for the induced mapping ^Si we have B^Si 2 Int(X ";Y #; I) and B B^Si for each B 2 Int(X ";Y #; I).
(c) J = Se / I with Se 2 LX X and for the induced mapping [Se we have A[Se 2 Ext(X ";Y #; I) and A A[Se for each A 2 Ext(X ";Y #; I).
Proof. (sketch, equivalence of (a) and (b)): We have Ext(X ";Y #; J) Ext(X ";Y #; I) iff there exists a matrix Si such that J = I . Si by [4, Theorem 7]. We have A"J = A"I.Si = (A"I )^Si for A 2 LX . Since A"I is any intent B in Int(X ";Y #; I) and A"J = B^Si 2 Int(X ";Y #; I) we obtain B^Si 2 Int(X ";Y #; I) for each B 2 Int(X ";Y #; I). By (16) and (14) we have that J(x; y) is equal to fxg"I ^Si (y) and with the property B B^Si we get I J proving that J is a block L-relation.
(a) The set of all block L-relations J of I is an L-closure system (i.e. it is closed under V and !).
(b) The set of all Se (from Theorem 9) is an L-interior system (i.e. it is closed under W and ).
(c) The set of all Si (from Theorem 9) is an L-interior system.
Proof. (a) Let Ji be block L-relations of I. Let J = Vi Ji and let B 2 Int(X ";Y #; J), hence B = A"J for some A 2 LX . By definition of "J and properties of residuated lattices we have
relation: (a) A[Se = A"I#J for any A 2 Ext(X ";Y #; I), B^Si = B#I"J for any B 2 Int(X ";Y #; I). (b) hA[Se ; A[Se \Se "I i 2 B(X ";Y #; J) and hB^Si _Si #I ; B^Si i 2 B(X ";Y #; J) for each hA; Bi 2 B(X ";Y #; I).
(c) A\Se "I = A"I^Si = A"J for any A 2 LX and B_Si #I = B#I[Se = B#J for any B 2 LY .
Let I 2 LX Y be an L-relation between X and Y and let J be its block relation. Denote by qJ a mapping qJ : B(X ";Y #; I) ! B(X ";Y #; I) defined by and denote by qJ a mapping qJ : B(X ";Y #; I) ! B(X ";Y #; I) defined by hA; BiqJ = hA[Se ; A[Se "I i hA; BiqJ = hB^Si #I ; B^Si i Notice, that different block L-relations J of an L-relation I induce different mappings qJ , qJ . In what follows we omit subscript in the notation qJ , qJ and write just q , q . Obviously, for each hA; Bi 2 B(X ";Y #; I) we have hA; Biq hA; Bi hA; Biq :
Now, we explain the relationship between block L-relations and compatible Ltolerances.
Theorem 11. (a) Let J I be a block L-relation. Then the mappings q ; q form an inflationary L-Galois connection on B(X ";Y #; I).
(b) For any inflationary L-Galois connection h f ; gi on B(X ";Y #; I) the L-relation
J(x; y) = f (hfxg"I#I ; fxg"I i)
hfyg#I ; fyg#I"I i ( = hfxg"I#I ; fxg"I i g(hfyg#I ; fyg#I"I i) ) is a block L-relation of hX ;Y; Ii such that its mappings q and q are equal to the mappings f and g, respectively. (25) (26) (27) Proof. (a) Since J is a block L-relation then A"J is an intent of hX ;Y; Ii and f (hA; Bi) 2 B(X ";Y #; I). Similarly for g.
The pair h"J ; #J i is an antitone L-Galois connection between LX and LY . Thus, for each A 2 LX and B 2 LY we have S(A; B#J ) = S(B; A"J ). Now, f (hA1; B1i) hA2; B2i = S(B2; A1"J ) = S(A1; B2#J ) = hA1; B1i g(hA2; B2i) proving h f ; gi is an isotone L-Galois connection. Since I J then A"I A"J and f is deflationary.
(b) First we need to show that each object intent of hX ;Y; Ji is an intent of hX ;Y; Ii and each attribute extent of hX ;Y; Ji is an extent of hX ;Y; Ii. We shall prove the part for extents. For each x 2 X we have fxg"J (y) = J(x; y) = S( fX (fxg"I#I ); fyg#I ) = ^ fX (fxg"I#I )(x0) ! fyg#I (x0) x02X = ^ fX (fxg"I#I )(x0) ! I(x0; y) = fX (fxg"I#I )"I (y)
x02X and fxg"J is an intent of hX ;Y; Ii. That proves that Ext(X ";Y #; J) Ext(X ";Y #; I); similarly can be shown that Int(X ";Y #; J) Int(X ";Y #; J). From properties of inflationary L-Galois connections and antitone Galois connections, we have fX (fxg"I#I )"I (y) fxg"I#I"I (y) = fxg"I (y); hence J I. The equality of q ; q and f ; g is immediate.
Theorem 11 and Theorem 8 generalize [7, Theorem 15]. The following theorem represents the main result of this paper.
Theorem 12. (a) Complete L-tolerances are in one-to-one correspondence with block L-relations.
(b) If complete L-tolerance and block L-relation J are in that correspondence, then B(X ";Y #; J) is isomorphic to the lattice of blocks of L-tolerance . Proof. The one-to-one correspondence follows from Theorem 11 and the fact that different block relation produce different mappings q , q . The isomorphism of B(X ";Y #; J) and the lattice of then follows from definition of q , q and their equality to mappings f ; g of .
In this section, we use results developed in the previous sections to factorize the Lconcept lattice B(X ";Y #; I) from Example 1.
Our aim is to reduce the size of the L-concept lattice B(X ";Y #; I) by
factorization by complete L-tolerances. First we take the approach from [
As an example, we present in Fig. 3 results we obtained for our formal context of movies, critics and ratings with values of the threshold a equal to 0:4 and 0:3.
0:6=c1; 0:6=c2; 0:4=c3; 0:6=c4; 0:6=c5 BV; 0:8=c5 0:8=c2 0:8=BV 0:8=c1 c5 0:8=c3
MD 0:6=BV; c2
We also tried a more sophisticated approach, based on [
Although a proper fuzzification of the results from [
For a formal L-context hX ;Y; Ii we select two L-sets X 0 2 LX and Y 0 2 LY and interpret them as L-sets of “important objects” and “important attributes”, respectively. Thus, for an object x 2 X , the value X 0(x) is the degree to which x is important, and similarly for attributes. Further we define two L-relations IX0 ; IY 0 2 LX Y by IX0 (x; y) = X 0(x) ! I(x; y);
IY 0 (x; y) = Y 0(y) ! I(x; y): As it can be easily seen from (15), Int(X ";Y #; IX0 ) Int(X ";Y #; I) and Ext(X ";Y #; IY 0 ) Ext(X ";Y #; I). Thus, the L-relations IX0 and IY 0 select some intents and extents of formal BV;0:6=c5 0:4=c3;0:6=c4 0:8=c2 LH;c1 Fig. 4. L-concept lattice of movies, critics and ratings (form Fig. 1), factorized with respect to an L-set of important objects X0 = f0:8=BV; LH; MD; TSSg and L-set of important attributes Y 0 = fc1; 0:8=c2; c3; c4; c5g (left) and with respect to an L-set of important objects X0 = fBV; LH; MD; TSSg and L-set of important attributes Y 0 = fc1; 0:6=c2; c3; c4; c5g (right). concepts from B(X ";Y #; I). These intents and extents are interpreted as “important”. Now, let JX0;Y 0 2 LX Y be the minimal (with respect to L-set inclusion) block L-relation of I such that both intent and extent of each formal L-concept from B(X ";Y #; JX0;Y 0 ) are important (JX0;Y 0 always exists because of Theorem 10(a); details are omitted). Using the results of this paper (Theorem 12) we obtain that JX0;Y 0 induces a complete L-tolerance on the L-concept lattice B(X ";Y #; I) and the corresponding factor completely lattice L-ordered set is isomorphic to the L-concept lattice B(X ";Y #; JX0;Y 0 ).
Note that this approach contains the approach from [
We apply the above considerations to our example. Suppose we consider the film BV less important than the other films (perhaps because we have not seen BV) and the critic c2 less important than the other critics (because we do not like his opinion on MD). More precisely, set X 0 = fa=BV; LH; MD; TSSg and Y 0 = fc1; b=c2; c3; c4; c5g, where a; b 2 L. In Fig. 4 we can see the resulting concept lattices in two cases: first a = b = 0:8 and second a = 1 and b = 0:6. 6
We presented a generalization of the theory of complete tolerances on complete lattices
and the relationship of block relations on formal contexts and complete tolerances on
the corresponding concept lattices [