Introduction and motivation

Let us have a group of schoolmates of a secondary grammar school and their studying results of ten subjects as it is shown in the table below. Names of subjects are in the table as abbreviations (Ma -Math, Sl -Slovak language, Ph -Physics, Ge -Geography, Bi -Biology, Gr -German, En -English, Ch -Chemistry, Ae -Aesthetics, Hi -History). Abbreviations of names of students are in the table. Ma Sl Ph Ge Bi Gr En Ch Ae Hi F Fred 1 1 1 3 2 1 2 2 1 2 J Joey 3 1 2 1 1 1 1 3 1 1 A Alice 3 2 3 1 1 1 1 3 2 2 N Nancy 4 2 4 3 2 2 1 2 3 2 M Mary 1 1 1 1 1 1 1 1 1 1 E Eve 1 1 1 1 1 1 1 1 1 1 L Lucy 1 3 1 2 2 2 2 1 2 2 D David 2 3 4 3 4 1 1 2 2 2 P Peter 2 1 2 1 1 2 2 3 1 2 T Tom 1 3 2 2 2 2 2 3 1 2

The table is a concrete example of fuzzy formal context. Students represent objects, subjects represent attributes and corresponding valuations represent values assigned to every object-attribute pair by fuzzy binary relation over the set {1, 2, 3, 4, 5} (1 -best, . . . , 5 -worst). Goal is to find groups of students similar by their studying results of all subjects, or to find subsets of subjects similar by results of all students. In other words to find pairs of classical subset of objects or attributes and fuzzy subset of attributes or objects. Similarity is determined by fuzzy subsets. Those pairs are called one-sided fuzzy concepts ( [1]).

The starting point of this paper is to define so-called proto-fuzzy concepts, triples made of a subset of objects, a subset of attributes and a value from the set of degrees of membership forming fuzzy binary relation, which is not exceeding for any object-attribute pair of cartesian product of object and attribute subsets meant above. Every element of the triple is "maximal" opposite to other two elements. Proto-fuzzy concepts can be taken as a "base structure unit" of one-sided fuzzy concepts. If values in the table are taken as columns tall as degree of membership of subsistent object-attribute pair to fuzzy binary relation, then proto-fuzzy concepts could be taken as a maximal "sub-blocks" of satisfying triples object-attribute-value of the 3D block representing fuzzy context. Examples of some proto-fuzzy concepts of the example will be shown in section 3. In the sense of simplicity of an idea "fuzzy" will be used instead of L-fuzzy. Lemma 1. For all l ∈ L the pair (↑ l , ↓ l ) forms a Galois connection between the power-set lattices P(O) and P(A).

Basic definitionsDefinition 4. Let O, A, r be a fuzzy context. A pair O, A is called an l- concept iff ↑ l (O) = A, and ↓ l (A) = O, hence the pair is a concept in a classical context O, A, R l , that R l = {(o, a) ∈ O × A : r(o, a) ≥ l}.

Context O, A, R l is called an l-cut. The set of all concepts in an l-cut will be denoted K l .

F • • • • • • • • • J • • • • • • • • A • • • • • • • N • • • • • • E • • • • • • • • • • M • • • • • • • • • • L • • • • • • • • • D • • • • • • P • • • • • • • • • T • • • • • • • •

In our example the l-cut means a look at the level of success for the value l. So the l-cut gives an Yes/No answer for the question: Is the result of each student in each subject at least of the value l? For example a concept O, A from K 2 represents the group O of students, that every subject of the set A is fulfilled at least in the value 2.

By exploring all l-cuts for such l ∈ L, it can be seen that some l-concepts are equal for different l ∈ L. But information that Eve and Mary are successful in all subjects for the value 2 is not complete and not as useful as information that they are successful for 1. This information is not complete, "closed".

Two interesting properties will be shown in following lemmas and theorems. It will be a continuation of the knowledge of the paper [3], where some properties of cuts was shown.

Lemma 2. Let l 1 , l 2 ∈ L that l 1 ≤ l 2 . ↑ l1 (O) ⊇↑ l2 (O) for every O ⊆ O and ↓ l1 (A) ⊇↓ l2 (A) for every A ⊆ A.

Proof. The proof will be shown for ↑. The proof for ↓ is likewise.

If

l 1 ≤ l 2 then {a ∈ A : (∀o ∈ O)r(o, a) ≥ l 1 } ⊇ {a ∈ A : (∀o ∈ O)r(o, a) ≥ l 2 }. Hence ↑ l1 (O) ⊇↑ l2 (O). Lemma 3. Let O ⊆ O, A ⊆ A and l 1 , l 2 ∈ L. Then ↑ l1 (O)∩ ↑ l2 (O) =↑ l1∨l2 (O) and ↓ l1 (A)∩ ↓ l2 (A) =↓ l1∨l2 (A). Proof. If a ∈↑ l1 (O)∩ ↑ l2 (O) then for all o ∈ O is r(o, a) ≥ l 1 and r(o, a) ≥ l 1 . It follows from above that for every o ∈ O is r(o, a) ≥ l 1 ∨ l 2 and so a ∈↑ l1∨l2 (O). Hence ↑ l1 (O)∩ ↑ l2 (O) ⊆↑ l1∨l2 (O). The lemma 6 implies that ↑ l1∨l2 (O) ⊆↑ l1 (O) and ↑ l1∨l2 (O) ⊆↑ l2 (O). It implies that ↑ l1∨l2 (O) ⊆↑ l1 (O)∩ ↑ l2 (O). From the both inclusions implies that ↑ l1∨l2 (O) =↑ l1 (O)∩ ↑ l2 (O). The proof for ↓ is likewise. Theorem 1. Let l 1 , l 2 ∈ L and O, A ∈ K l1 ∩ K l2 . Then for all l ∈ L, if l 1 ≤ l ≤ l 2 then O, A ∈ K l .

Proof. The lemma 6 and

O, A ∈ K l1 ∩ K l2 implies that A =↑ l1 (O) ⊇↑ l (O) ⊇↑ l2 (O) = A, O =↓ l1 (A) ⊇↓ l (A) ⊇↓ l2 (A) = O Hence ↑ l (O) = A and ↓ l (A) = O, which implies O, A ∈ K l . Theorem 2. Let l 1 , l 2 ∈ L and O, A ∈ K l1 ∩ K l2 . Then O, A ∈ K l1∨l2 .

Proof. The lemma 7 implies

↑ l1∨l2 (O) =↑ l1 (O)∩ ↑ l2 (O) = A ∩ A = A,, ↓ l1∨l2 (A) =↓ l1 (A)∩ ↓ l2 (A) = O ∩ O = O. Hence O, A ∈ K l1∨l2 .

3 Proto-fuzzy concepts and their usage

Definition 5. Triples O, A, l ∈ P(O) × P(A) × L such that O, A ∈ k∈L K k and l = sup{k ∈ L : O, A ∈ K k } will be called proto-fuzzy concepts.

The set of all proto-fuzzy concepts will be denoted K P .

For our example will proto-fuzzy concept O, A, l means the group of students O, whose best common result of all subjects from the set A is l. In the following tables are some proto-fuzzy concepts of our example. {F, J, A, N, M, E, L, D, P, T} {Sl, Ge, Gr, En, Ch, Ae, Hi} 3 {F, J, A, P, E, M} {Sl, Bi, Ae , Gr, En , Hi} 2

{F, M, E, L} {Ma, Ph}1

The set of all proto-fuzzy concepts will be used for creating one-sided fuzzy concepts with help of mappings defined below. Mappings will determine which side will be fuzzy.

⇑: 2 O → L A , ⇓: L A → 2 O

in the following way: For every subset O of objects and for every fuzzy-subsets of attributes put

⇑ (O)(a) = sup{l ∈ L : (∃ A, l ∈ K P O )a ∈ A} ⇓ ( A) = {O ⊆ O : (∀a ∈ A)(∃ A, l ∈ K P O )a ∈ A & l ≥ A(a)}.Then O =↓ l (A) =↓ l (↑ l (O))

and from the fact that for every l ∈ L the pair (↑ l , ↓ l ) forms a Galois connection, it implies that

↓ l (↑ l (O)) ⊇ O and hence O, A ∈ K l . If k = sup{m ∈ L : O, A ∈ K m } the theorem 9 implies that O, A ∈ K k and so O, A, k ∈ K P . Lemma 5. Let l ∈ L, O 1 , O 2 ⊆ O and A 1 , l 1 ∈ K P O1 , A 2 , l 2 ∈ K P O2 that A 1 ∩ A 2 = ∅ and l 1 ∧ l 2 ≥ l. Then exists A, k ∈ K P O1∪O2 that A ⊇ A 1 ∩ A 2 and k ≥ l. Proof. A 1 , l 1 ∈ K P O1 it means that (∀o ∈ O 1 )(∀a ∈ A 1 )r(o, a) ≥ l 1 . A 2 , l 2 ∈ K P O2 it means that (∀o ∈ O 2 )(∀a ∈ A 2 )r(o, a) ≥ l 2 . Hence (∀o ∈ O 1 ∪ O 2 )(∀a ∈ A 1 ∩ A 2 )r(o, a) ≥ l 1 ∧ l 2 ≥ l.

The lemma 13 implies that

(∃O ⊇ O 1 ∪ O 2 )(∃A ⊇ A 1 ∩ A 2 )(∃k ∈ L : k ≥ l) O, A, k ∈ K P hence A, k ∈ K P O1∪O2 . Lemma 6. Let O ⊆ O, A 1 , l 1 , A 2 , l 2 ∈ K P O such that A 1 ∩ A 2 = ∅. Then there exist A ⊆ A and l ≥ l 1 ∨ l 2 such that A, l ∈ K P O .

Proof. For all o ∈ O and for all a

∈ A 1 ∩ A 2 is r(o, a) ≥ l 1 and r(o, a) ≥ l 2 . Hence r(o, a) ≥ l 1 ∨ l 2 .

From above and lemma 13 implies that there exist

(∃B ⊇ O)(∃A ⊇ A 1 ∩ A 2 )(∃l ∈ L : l ≥ l 1 ∨ l 2 ) B, A, k ∈ K P . Hence A, l ∈ K P O .

Lemma 7. Let O 1 , O 2 be an arbitrary subsets of the set of objects such that

O 1 ⊆ O 2 . Then K P O1 ⊇ K P O2 . Proof. Because of O 1 ⊆ O 2 is { A 1 , l 1 ∈ P(A) × L : (∃B 1 ⊇ O 1 ) B 1 , A 1 , l 1 ∈ K P } ⊇ ⊇ { A 2 , l 2 ∈ P(A) × L : (∃B 2 ⊇ O 2 ) B 2 , A 2 , l ∈ K P }. Hence K P O1 ⊇ K P O2 .

Theorem 3. The pair of mappings (⇑, ⇓) forms a Galois connection between the power-set lattice P(O) and the fuzzy power-set lattice F(A).

Proof. For every set O, the subset of the set of objects and the fuzzy set A, the fuzzy-subset of the set of attributes, have to be proven that O is the subset of ⇓ ( A) if, and only if A is the fuzzy-subset of ⇑ (O).

⇒ O ⊆⇓ ( A) = {B ⊆ O : (∀b ∈ A)(∃ A, l ∈ K P B )b ∈ A & l ≥ A(b)}.

Let a ∈ A be an arbitrary attribute. The lemma 14 implies that there exists

A a ⊆ A, l a ∈ L such that a ∈ A a , l a ≥ A(a) and A a , l a ∈ K P ⇓( A) . O ⊆⇓ ( A) implies that K P O ⊇ K P ⇓( A) . Hence A a , l a ∈ K P O . So A(a) ≤ l a ≤ sup{l ∈ L : (∃ A, l ∈ K P O )a ∈ A} =⇑ (O)(a).

Because of arbitrarity of attribute a and from unequality above implies that A is the fuzzy-subset of ⇑ (O). ⇐ Let a ∈ A be an arbitrary attribute. Denote

l a =⇑ (O)(a) = sup{l ∈ L : (∃ A, l ∈ K P O )a ∈ A}.

The proposition implies that for every a ∈ A(a) ≤ l a . The lemma 15 implies that there exists A a ⊆ A such that A a , l a ∈ K P O , and that implies

O ∈ {B ⊆ O : (∀b ∈ A)(∃ A, l ∈ K P B )a ∈ A & l ≥ A(b)} hence O ⊆ {B ⊆ O : (∀b ∈ A)(∃ A, l ∈ K P B )a ∈ A & l ≥ A(b)} =⇓ ( A).

So the set O is subset of ⇓ ( A).

For the case of object fuzzy side will be used mappings:

⇑: 2 A → L O , ⇓: L O → 2 A .

Let O be a fuzzy subset of objects and A ⊆ A is subset of attributes.

⇑ (A)(o) = sup{l ∈ L : (∃ O, l ∈ K P A )o ∈ O} ⇓ ( O) = {T ⊆ A : (∀o ∈ O)(∃ O, l ∈ K P T )o ∈ O & l ≥ O(a)},

where

K P A = { O, l : (∃T ⊇ A) O, T, l ∈ K P }. Example 1.

For example take the fuzzy-subset of the set of attributes,

A = {(Ma,1), (Sl,3), (Ph,1), (Ge,3), (Bi,4), (Gr,2), (En,2), (Ch,2), (Ae,4), (Hi,4)}.

In the table below are some proto-fuzzy concepts which contains students whose results satisfy to A. Hence ⇓ ( A) ={F,L,M,E}. Elements of

K P ⇓( A)

are shown in the next table . Hence

⇑ (⇓ ( A)) = = {(Ma,1

),(Sj,3),(Ph,1),(Ge,3),(Bi,2),(Gr,2),(En,2),(Ch,2),(Ae,2),(Hi,2)} .

Retrieval of proto-fuzzy concepts

Proto-fuzzy concepts will be retrieved with a help of cuts and "pessimistic sights" to object-value or attribute-value plains.

Definition 8. Define new binary relations

R A = {(o, l) ∈ O × L : (∀a ∈ A)r(o, a) ≥ l} and R O = {(a, l) ∈ A × L : (∀o ∈ O)r(o, a) ≥ l}.

The formal context O, L, R A will be called object-value sight and the formal context A, L, R O will be called attribute-value sight.

Table 5. Object-value and attribute-value sight

1 2 3 4 5 Fred • • • Joey • • • Alice • • • Nancy • • Mary • • • • • Eve • • • • • Lucy • • • David • • Peter • • • Tom • • • 1 2 3 4 5 Math • • Slovak language • • • Physics • • Geography • • • Biology • • German language • • • • English language • • • • Chemistry • • • Aesthetics • • • History • • • • Definition 9. Define new mappings ↑ A : 2 O → L and ↓ A : L → 2 O , ↑ O : 2 A → L and ↓ O : L → 2 A .

For every O ⊆ O, A ⊆ A and l ∈ L put Proof. The proof will be shown only for first pair. The proof for second pair is likewise.

↑ A (O) = inf{sup{l ∈ L : (o, l) ∈ R A } : o ∈ O} ↓ A (l) = {o ∈ O : (o, l) ∈ R A } ↑ O (A) = inf{sup{l ∈ L : (a, l) ∈ R O } : a ∈ A} ↓ O (l) = {a ∈ A : (a, l) ∈ R O }.1. Let O 1 ⊆ O 2 ⊆ O. It follows from an inclusion above that {sup{l ∈ L : (o, l) ∈ R A } : o ∈ O 1 } ⊆ ⊆ {sup{l ∈ L : (o, l) ∈ R A } : o ∈ O 2 }

and from a properties of infimum

inf{sup{l ∈ L : (o, l) ∈ R A } : o ∈ O 1 } ≥ ≥ inf{sup{l ∈ L : (o, l) ∈ R A } : o ∈ O 2 }. Hence ↑ A (O 1 ) ≥↑ A (O 2 ). 2. Let l 1 , l 2 ∈ L. If l 1 ≤ l 2 then {o ∈ O : (o, l 1 ) ∈ R A } ⊇ {o ∈ O : (o, l 2 ) ∈ R A }. Hence ↓ A (l 1 ) ⊇↓ A (l 2 ).↓ A (↑ A (O)) ⊇↓ A (s o ) = {b ∈ O : (b, s o ) ∈ R A }.

Arbitrarity of o implies that↓ A (↑ A (O)) ⊇ o∈O {b ∈ O : (b, s o ) ∈ R A } ⊇ O.• • • • • • J • • • • • A • • • • N • • • L • • • • • • D • • • P • • • • • • T • • • • •

Because of the convexity of l-concepts, we can omit Eve and Mary from the set of students for auxiliary subcontext of 2-cut. And for input of theorem 23 for degree 2 can be used proto-fuzzy concepts {M,E}, A, 1 , O, {Gr,En,Hi}, 2 ∈ K P .

• • • • • J • • • • • • • A • • • • N • L • • • D • • P • • • • T • •

Conclusion

Conceptual scaling and theory of triadic contexts will be the object of our future work and study. We will try to algoritmize outline process. We are grateful for precious comments of our colleagues RNDr. Peter Eliaš PhD. and RNDr. Jozef Pócs.

Paper was created with support of grant 1/3129/06 Slovak grant agency VEGA.