In this work, we introduce and study the composition of two L-fuzzy contexts that share the same attribute set. Besides studying its properties, this composition allows to establish relations between the sets of objects associated to both L-fuzzy contexts. We also de ne, as a particular case, the composition of an L-fuzzy context with itself. In all the cases, we show some examples that illustrate the results.
In some situations we have information that relates two sets X and Z to the same set Y and we want to know if these relations allow us to establish connections between X and Z. In the present work we will try to deal with the study of this problem using as tool the L-fuzzy Concepts Theory.
The Formal Concept Analysis developed by Wille ([
The set of the concepts derived from a context (X; Y; R) is a complete lattice and it is usually represented by a line diagram.
In some previous works ([
B2(x) = inf fI(B(y); R(x; y))g y2Y where I is a fuzzy implication operator de ned in (L; ), I : L L ! L; which is decreasing in its rst argument, and, A1 represents, as a fuzzy set, the attributes related to the objects of A and B2 the objects related to the attributes of B.
The information of the context is visualized by means of the L-fuzzy concepts which are pairs (A; A1) 2 (LX ; LY ) with A 2 x(') the set of xed points of the operator ', being this one de ned by the derivation operators 1 and 2 mentioned above as '(A) = (A1)2 = A12. These pairs, whose rst and second components are the extension and the intension respectively, represent, as a fuzzy set, the set of objects that share some attributes.
The set L = f(A; A1) : A 2 x(')g with the order relation de ned as: (A; A1); (C; C1) 2 L; (A; A1) (C; C1) if A
C
(or equiv. C1 A1) is a complete lattice that is said to be the L-fuzzy concept
lattice ([
On the other hand, given A 2 LX ; (or B 2 LY ) we can obtain the derived L-fuzzy concept applying the de ned derivation operators. In the case of the use of a residuated implication operator (as it holds in this work), the associated L-fuzzy concept is (A12; A1) (or (B2; B21)):
Other extensions of the Formal Concept Analysis to the fuzzy area are in
[
The composition of formal contexts allows to establish relations between the elements of two sets of objects that share the same attribute set.
De nition 1. Let (X; Y; R1) and (Z; Y; R2) be two formal contexts, the composed formal context is de ned as the context (X; Z; R1 ? R2), where 8(x; z) 2 X Z:
R1 ? R2(x; z) = (1 if R2(z; y) = 1; 8y such that R1(x; y) = 1
0 in other case That is, the object x is related to z in the composed context if z shares all the attributes of x in the original contexts.
Proposition 1. The relation of the composed context, R1 ? R2, can also be de ned as:
R1 ? R2(x; z) = minfmaxfR10(x; y); R2(z; y)gg y2Y where R10 is the negation of the relation R1, that is, R10(x; y) = (R1(x; y))0 8(x; y) 2 X Y .
This property will be helpful in the following sections.
Remark 1. Given the formal contexts (X; Y; R1) and (Z; Y; R2), the relation of the composed context R1 ? R2 is not necessarily the opposed of the relation R2 ? R1, that is, in general,
There exists (x; z) 2 X
Z such that R1 ? R2(x; z) 6= R2 ? R1(z; x) Example 1. Let us consider the formal contexts (X; Y; R1) and (Z; Y; R2), where X = fx1; x2; x3g, Y = fy1; y2; y3; y4; y5g, Z = fz1; z2; z3; z4g, and the respective relations are the following ones:
If we calculate the composition of the contexts de ned above in the two possible orders, then the obtained relations are: and, as can be seen, (R1 ? R2)op 6= R2 ? R1.
This property will be helpful in the following sections. 2.1
Particular case: when a formal context is composed with itself Let us analyze a particular case where some interesting results are obtained. Proposition 2. Let (X; Y; R) be a formal context. If (X; Y; R) is composed with itself, then the obtained context is (X; X; R ? R) where the sets of objects and attributes are coincident and the relation R ? R is a binary relation de ned on X as follows:
R ? R(x1; x2) = minfmaxfR0(x1; y); R(x2; y)gg 8(x1; x2) 2 X y2Y The composition of this context with itself is the context (X; X; R ? R), and relation is given by the table:
Remark 2. The object x1 is related to attribute x2 in the composed context, if in the original context the object x2 has at least the same attributes than the object x1.
Example 2. Returning to the formal context (X; Y; R) that we studied in the previous example, where the relation R was: Proposition 3. The relation R ? R obtained by the composition of the formal context (X; Y; R) with itself is a preorder relation de ned on the object set X. Proof. As a consequence of the de nition, it is immediate to prove that: 1. The relation R ? R is re exive. 2. The relation R ? R is transitive. tu Remark 3. It is a simple veri cation to see that: { The relation R ? R is not, in general, a symmetric relation. To be symmetric it is necessary that whenever an object x2 in the original context (X; Y; R) has all the attributes of another object x1, both objects have the same set of attributes. { The relation R ? R is not antisymmetric either. Therefore, R ? R is not, in general, an order relation. 3
Extension to the L-fuzzy context case The expression given in proposition 1 can be generalized to the fuzzy case substituting the maximum operator by a t-conorm S and taking a strong negation 0. In this way, we can de ne the compositions of two L-fuzzy contexts as follows: De nition 2. Let (L; X; Y; R1) and (L; Z; Y; R2) be two L-fuzzy contexts, we de ne the composed L-fuzzy context (L; X; Z; R1 ? R2), where:
R1 ? R2(x; z) = inf fS(R10(x; y); R2(z; y))g y2Y
If we remind the de nition of a fuzzy S-implication, the previous one can be expressed in this way: De nition 3. Let (L; X; Y; R1) and (L; Z; Y; R2) be two L-fuzzy contexts, and I an S-implication operator. We de ne the composed L-fuzzy context (L; X; Z; R1 ? R2), where:
R1 ? R2(x; z) = inf fI(R1(x; y); R2(z; y))g
y2Y
Remark 4. If we remind the de nition of the triangle subproduct operator / given
by [
R1 ?I R2 = R1 / (R2)op As can be observed, also in this case a similar result to the crisp case is obtained. Proposition 4. Let (L; X; Y; R1) and (L; Z; Y; R2) be two L-fuzzy contexts. Then, the relation of the composed L-fuzzy context (L; X; Z; R1?I R2) is not, in general, the opposite of the relation of the composed L-fuzzy context (L; Z; X; R2 ?I R1).
(R1 ?I R2)op 6= R2 ?I R1 That is, if we change the order of the composition, the obtained relation between the elements of X and Z is di erent.
Proof. Given two L-fuzzy contexts (L; X; Y; R1) and (L; Z; Y; R2), and a fuzzy implication operator I, the relation of the composed L-fuzzy context (L; X; Z; R1 ?I R2) is: As, in general, given a fuzzy implication I(a; b) 6= I(b; a), then these relations are not opposed. tu Example 3. We have a company of temporary work in which we want to analyze the suitability of some candidates to obtain some o ered employments. The company knows the requirements of knowledge to occupy each one of the positions, represented by means of the L-fuzzy context (L; X; Y; R1), where the set of objects X is the set of employments, the attributes Y the necessary knowledge, and the relation among them appears in Table 1 with values in the chain L=f0, 0.1, 0.2, . . . , 1g.
On the other hand, we have the knowledge of some candidates for these positions, represented by the L-fuzzy context (L; Z; Y; R2) in which the objects are the di erent candidates to occupy the jobs, the attributes the necessary knowledge and the relation among them is given by Table 2.
A candidate will be suitable to obtain a job if he owns all the knowledge required in this position. Therefore, to analyze what candidate is adapted for each job, we would use the composed L-fuzzy context (L; X; Z; R1 ? R2). The relation of this composed context, calculated using the Lukasiewicz implication operator, is the represented in Table 3.
To obtain the information of this L-fuzzy context we will use the ordinary tools of the L-fuzzy Concept Theory to analyze the associated L-fuzzy concepts. Thus, for example, if we want to nd the best candidate to occupy the job of waiter, we take the set: fdomestic helper=0; waiter=1; accountant=0; car salesman=0g and we obtain the derived L-fuzzy concept, whose intension is:
fC1=0:9; C2=1; C3=0:6; C4=0:8; C5=0:4g
If we look at the attributes with the highest membership degree, we can deduce that the most suitable candidate for the job of waiter is C2, followed by C1 and C4.
If, for instance, we want to nd the best person to be accountant in a restaurant that also could work as a waiter, we take the set
fdomestic helper=0; waiter=1; accountant=1; car salesman=0g and the derived L-fuzzy concept is
fC1=0:6; C2=0:3; C3=0:1; C4=0:4; C5=0:4g where we can see that the most suitable candidate is C2.
On the other hand, if our interest is to analyze which of the jobs is the most suitable for each candidate, we do the composition in the contrary order, obtaining the L-fuzzy context (L; Z; X; R2 ? R1), where the composed relation is represented in Table 4.
We can see in this example that both compositions are di erent: A candidate can be the best to occupy a concrete job, but that job need not be the most appropriate for this candidate.
The following result will be of interest to study the L-fuzzy concepts associated to the objects of the composed L-fuzzy context.
Before to proceed with the proposition, we are going to introduce a new notation: If the subscripts point out the derivation operators and the superscripts the L-fuzzy contexts where they are applied, then A1? is the derived set from A obtained in the composed L-fuzzy context, A11 is the derived set obtained in the L-fuzzy context (L; X; Y; R1), and (A11 )22 the derived set of the last one in the L-fuzzy context (L; Z; Y; R2) Proposition 5. If the implication operator I is residuated and we consider the set:
A(x) = (1 if x = xi
0 in other case then, the intension of the L-fuzzy concept obtained in the composed L-fuzzy context (L; X; Z; R1 ?I R2) from the set A, is equal to the extension of the L-fuzzy concept obtained in (L; Z; Y; R2) from the intension of the L-fuzzy concept obtained in (L; X; Y; R1) from A. That is, we obtain the same fuzzy set Z applying the derivation operators twice (once in each one of the contexts that make up the composition), or once in the composed context.
Moreover, it is verify that: 8z 2 Z;
A1? (z) = (A11 )22 (z) = R1 ?I R2(xi; z) That is, the membership degrees obtained are the values of the row of R1 ?I R2 that corresponds to the object xi.
(1 if x = xi Proof. Let be A(x) =
0 in other case obtained from A in the context (L; X; Y; R1) is the L-fuzzy subset of Y : , the intension of the L-fuzzy concept A11 (y) = inf fI(A(x); R1(x; y))g; x2X 8y 2 Y: As the implication I is residuated, 8a 2 L it is veri ed that I(0; a) = 1 and I(1; a) = a, thus,
A11 (y) = R1(xi; y); Taking now the set A11 , we obtain the derived L-fuzzy concept in the L-fuzzy context (L; Z; Y; R2), the extension of which is: (A11 )22 (z) = inf fI(A11 (y); R2(z; y))g =
y2Y = inf fI(R1(xi; y); R2(z; y))g = R1 ?I R2(xi; z); y2Y 8z 2 Z:
On the other hand, the intension of the obtained L-fuzzy concept in the composed L-fuzzy context from A is:
A1? (z) = inf fI(A(x); R1 ?I R2(x; y))g = R1 ?I R2(xi; z); x2X 8z 2 Z: tu Example 4. If we come back to example 3, we have analyzed which candidate is the most suitable for the job of waiter.
To do this, in the L-fuzzy context (L; X; Z; R1 ? R2) (see Table 3) we have taken the set
A = fdomestic helper=0; waiter=1; accountant=0; car salesman=0g and we have calculated the closed L-fuzzy concept, where the fuzzy intension is:
A1? = fC1=0:9; C2=1; C3=0:6; C4=0:8; C5=0:4g And here, if we look at those attributes whose membership degrees stand out from the others, we deduce that the most suitable candidates to be good waiters were, C2, C1 and C4, in this order.
The same result is obtained if we take the L-fuzzy context (L; X; Y; R1) (see Table1) and we calculate the L-fuzzy concept from A, which intension is:
A11 = fcomputer science=0; accounting=0:4; mechanics=0; cooking=0:7g And, from this fuzzy set we obtain in the L-fuzzy context (L; Z; Y; R2) (see Table2) the derived L-fuzzy concept the extension of which is:
(A11 )22 = fC1=0:9; C2=1; C3=0:6; C4=0:8; C5=0:4g As can be seen, the result is the same that the obtained in the composed L-fuzzy context. 3.2
Composition of an L-fuzzy context with itself The composition of an L-fuzzy context (L; X; Y; R) with itself will allow us to set up some relationships between the elements of the object set X. Proposition 6. If I is a residuated implication associated with a left continuous t-conorm T , then the relation R?I R that results of the composition of (L; X; Y; R) with itself, associated with the implication I, constitutes a fuzzy preorder relation de ned in the object set X.
Proof. 1. First, we prove that it is a re exive relation, that is, the relation veri es:
8x 2 X; R ?I R(x; x) = 1: By the de nition of the composition associated with an implication operator, we have 8x 2 X;
R ?I R(x; x) = inf fI(R(x; y); R(x; y))g;
y2Y and, as any residuated implication veri es that I(a; a) = 1; 8a 2 L, then 8x 2 X;
R ?I R(x; x) = 1: 2. To see that R ?I R is a T -transitive relation, we have to prove that 8x; t; z 2 X;
T (R ?I R(x; t); R ?I R(t; z))
R ?I R(x; z); that is, the following inequality must be veri ed:
T inf fI(R(x; ); R(t; ))g; inf fI(R(t; ); R(z; ))g 2Y 2Y inf fI(R(x; ); R(z; ))g: 2Y By the monotony of the t-norm, we have:
T inf 2Y inf fI(R(x; ); R(t; ))g; inf fI(R(t; ); R(z; ))g 2Y 2Y
T
I(R(x; ); R(t; )); inf fI(R(t; ); R(z; ))g
2Y
i2nYf fT (I(R(x; ); R(t; )); I(R(t; ); R(z; )))g :
As the used t-norm T is left-continuous, we know that [
I(a; c); and it is veri ed that:
T inf fI(R(x; ); R(t; ))g; inf fI(R(t; ); R(z; ))g 2Y 2Y inf fI(R(x; ); R(z; ))g: 2Y tu Remark 5. The relation R ?I R is neither symmetric nor antisymmetric and then, is neither an equivalence nor an order relation. For instance, if we take the relation R given by the table: then the relation R ?I R associated with the Lukasiewicz implication operator is:
R ?I R x1 x2
x3 and, as can be seen, is neither a symmetric nor an antisymmetric relation. Remark 6. If we are using a non residuated implication operator, not always a fuzzy preorder relation is obtained. For instance, if we take the previous relation R and we do the composition R ?I R associated with the Kleene-Dienes implication (that does not verify I(x; x) = 1), then we obtain the following relation: R ?I R x1 x2
x3 that is neither a re exive nor a fuzzy preorder relation.
The application of this composition can be very interesting in social or work relations as we can see next: Example 5. There are four di erent manufacture processes in a factory and we want to organize the workers so that each of them is subordinate of another one if its capacity to carry out each one of the processes of manufacture is smaller.
To model this problem, we are going to take the L-fuzzy context (L; X; Y; R), where the set of objects X is formed by the workers fO1; O2; O3; O4; O5g, the attributes are the di erent manufacture processes fP1; P2; P3; P4g, and the relation R represents the capacity of each one of the workers to carry out each one of the processes, in a scale of 0 to 1 (See Table 5).
The L-fuzzy context that results of the composition of this context with itself allow us to de ne relations boss-subordinate between the workers so that the relation R ? R(x; y) of the compound context (associated with the Lukasiewicz implication) gives the degree in which the worker x is subordinate of the worker y. (See Table 6).
This will allow us, for example, to choose bosses in the group watching the columns of the obtained relation: In this case, we could choose as bosses of the workers to O2 and O5 because both have as subordinate O3 and O4 and the subordination degrees are the biggest values of the columns. 4
This work constitutes the rst approach to the problem of composition of Lfuzzy contexts. In future works we will use these results in the resolution of other problems that seem interesting to us:
- First, this composition will be useful to study the chained L-fuzzy contexts, that is, to nd relations between two de ned contexts where the set of attributes of the rst context is the same that the set of objects of the second one.
- On the other hand, we think that it will be useful to de ne the composition of L-fuzzy contexts in the interval-valued case in order to study certain situations.
This work has been partially supported by the Research Group \Intelligent Systems and Energy (SI+E)" of the Basque Government, under Grant IT519-10.