=Paper=
{{Paper
|id=Vol-433/paper-11
|storemode=property
|title=Proto-fuzzy Concepts, their Retrieval and Usage
|pdfUrl=https://ceur-ws.org/Vol-433/paper7.pdf
|volume=Vol-433
}}
==Proto-fuzzy Concepts, their Retrieval and Usage==
https://ceur-ws.org/Vol-433/paper7.pdf
Proto-fuzzy Concepts, their Retrieval and Usage
Ondrej Krı́dlo and Stanislav Krajči
University of Pavol Jozef Šafárik, Košice, Slovakia
Abstract. The aim of this paper is to define so-called proto-fuzzy con-
cepts, as a base for generating different types of one-sided fuzzy concept
lattices. Fuzzy formal context is a triple of a set of objects, a set of at-
tributes and a fuzzy binary relation over a complete residuated lattice,
which determines the degree of membership of each attribute to each
object. A proto-fuzzy concept is a triple of a subset of objects, a subset
of attributes and a value as the best common degree of membership of
all pairs of objects and attributes from the above-mentioned sets to the
fuzzy binary relation. Then the proto-fuzzy concepts will be found with
a help of cuts and projections to the object-values or attribute-values
plains of our fuzzy-context.
1 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.
Table 1. Example of fuzzy formal context.
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
c Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 83–95,
ISBN 978–80–244–2111–7, Palacký University, Olomouc, 2008.
84 Ondrej Krı́dlo, Stanislav Krajči
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 exceed-
ing 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 de-
gree of membership of subsistent object–attribute pair to fuzzy binary relation,
then proto-fuzzy concepts could be taken as a maximal “sub-blocks” of satis-
fying 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.
2 Basic definitions
Definition 1. A formal context is a triple hO, A, Ri consists of two sets O, the
set of objects, and A, the set of attributes, and a relation R ⊆ O × A.
Definition 2. A fuzzy formal context is a triple hO, A, ri consists of two sets
O, the set of objects, and A, the set of attributes, and r is fuzzy subset of
O × A, mapping from O × A to L, where L is a lattice.
In the sense of simplicity of an idea “fuzzy” will be used instead of L-fuzzy.
Definition 3. For every l ∈ L define mappings ↑l : P(O) → P(A) and ↓l :
P(A) → P(O): For every subset O ⊆ O put
↑l (O) = {a ∈ A : (∀o ∈ O)r(o, a) ≥ l}
and for all A ⊆ A put
↓l (A) = {o ∈ O : (∀a ∈ A)r(o, a) ≥ l}.
Lemma 1. For all l ∈ L the pair (↑l , ↓l ) forms a Galois connection between the
power-set lattices P(O) and P(A).
Definition 4. Let hO, A, ri be a fuzzy context. A pair hO, Ai is called an l-
concept iff ↑l (O) = A, and ↓l (A) = O, hence the pair is a concept in a classical
context hO, A, Rl i, that
Rl = {(o, a) ∈ O × A : r(o, a) ≥ l}.
Context hO, A, Rl i is called an l-cut. The set of all concepts in an l-cut will be
denoted Kl .
Proto-fuzzy Concepts, their Retrieval and Usage 85
Table 2. 2-cut.
2 Ma Sl Ph Ge Bi Gr En Ch Ae Hi
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 hO, Ai
from K2 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 l1 , l2 ∈ L that l1 ≤ l2 . ↑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 l1 ≤ l2 then
{a ∈ A : (∀o ∈ O)r(o, a) ≥ l1 } ⊇ {a ∈ A : (∀o ∈ O)r(o, a) ≥ l2 }.
Hence ↑l1 (O) ⊇↑l2 (O). t
u
Lemma 3. Let O ⊆ O, A ⊆ A and l1 , l2 ∈ 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) ≥ l1 and r(o, a) ≥ l1 . It
follows from above that for every o ∈ O is r(o, a) ≥ l1 ∨ l2 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. t
u
86 Ondrej Krı́dlo, Stanislav Krajči
Theorem 1. Let l1 , l2 ∈ L and hO, Ai ∈ Kl1 ∩ Kl2 . Then for all l ∈ L, if
l1 ≤ l ≤ l2 then hO, Ai ∈ Kl .
Proof. The lemma 6 and hO, Ai ∈ Kl1 ∩ Kl2 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 hO, Ai ∈ Kl . t
u
Theorem 2. Let l1 , l2 ∈ L and hO, Ai ∈ Kl1 ∩ Kl2 . Then hO, Ai ∈ Kl1 ∨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 hO, Ai ∈ Kl1 ∨l2 . t
u
3 Proto-fuzzy concepts and their usage
S 5. Triples hO, A, li ∈ P(O) × P(A) × L such that
Definition
hO, Ai ∈ k∈L Kk and l = sup{k ∈ L : hO, Ai ∈ Kk } will be called proto-fuzzy
concepts. The set of all proto-fuzzy concepts will be denoted KP .
For our example will proto-fuzzy concept hO, A, li means the group of stu-
dents 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} {F, J, A, P, E, M} {F, M, E, L}
{Sl, Ge, Gr, En, Ch, Ae, Hi} {Sl, Bi, Ae , Gr, En , Hi} {Ma, Ph}
3 2 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.
Definition 6. Let O ⊆ O be an arbitrary set of objects. The set
P
KO = {hA, li ∈ P(A) × L : (∃B ⊇ O)hB, A, li ∈ KP }
will be called the contraction of the set of proto-fuzzy concepts subsistent to the
set O.
Definition 7. Define mappings
⇑: 2O → LA ,
⇓: LA → 2O
in the following way: For every subset O of objects and for every fuzzy-subsets
of attributes put
P
⇑ (O)(a) = sup{l ∈ L : (∃hA, li ∈ KO )a ∈ A}
[
P
⇓ (A)
e = {O ⊆ O : (∀a ∈ A)(∃hA, li ∈ KO )a ∈ A & l ≥ A(a)}.
e
Proto-fuzzy Concepts, their Retrieval and Usage 87
Lemma 4. Let O and A are arbitrary subsets of objects and attributes respec-
tively, and l be an arbitrary value of L such that for every object o of the set
O and for every attribute a of the set A, R(o, a) ≥ l. Then there exist subsets
O ⊇ O, A ⊇ A and value k ∈ L such that k ≥ l and hO, A, ki ∈ KP .
Proof. It is given that (∀o ∈ O)(∀a ∈ A)r(o, a) ≥ l. Take
A =↑l (O) = {a ∈ A : (∀o ∈ O)r(o, a) ≥ l} ⊇ 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 hO, Ai ∈ Kl . If
k = sup{m ∈ L : hO, Ai ∈ Km }
the theorem 9 implies that hO, Ai ∈ Kk and so
hO, A, ki ∈ KP .
t
u
P P
Lemma 5. Let l ∈ L, O1 , O2 ⊆ O and hA1 , l1 i ∈ KO 1
, hA2 , l2 i ∈ KO 2
that
P
A1 ∩ A2 6= ∅ and l1 ∧ l2 ≥ l. Then exists hA, ki ∈ KO1 ∪O2 that A ⊇ A1 ∩ A2 and
k ≥ l.
P
Proof. hA1 , l1 i ∈ KO 1
it means that
(∀o ∈ O1 )(∀a ∈ A1 )r(o, a) ≥ l1 .
P
hA2 , l2 i ∈ KO 2
it means that
(∀o ∈ O2 )(∀a ∈ A2 )r(o, a) ≥ l2 .
Hence
(∀o ∈ O1 ∪ O2 )(∀a ∈ A1 ∩ A2 )r(o, a) ≥ l1 ∧ l2 ≥ l.
The lemma 13 implies that
(∃O ⊇ O1 ∪ O2 )(∃A ⊇ A1 ∩ A2 )(∃k ∈ L : k ≥ l)hO, A, ki ∈ KP
hence
P
hA, ki ∈ KO 1 ∪O2
.
t
u
P
Lemma 6. Let O ⊆ O, hA1 , l1 i, hA2 , l2 i ∈ KO such that A1 ∩ A2 6= ∅. Then
P
there exist A ⊆ A and l ≥ l1 ∨ l2 such that hA, li ∈ KO .
88 Ondrej Krı́dlo, Stanislav Krajči
Proof. For all o ∈ O and for all a ∈ A1 ∩ A2 is
r(o, a) ≥ l1 and r(o, a) ≥ l2 .
Hence
r(o, a) ≥ l1 ∨ l2 .
From above and lemma 13 implies that there exist
(∃B ⊇ O)(∃A ⊇ A1 ∩ A2 )(∃l ∈ L : l ≥ l1 ∨ l2 )hB, A, ki ∈ KP .
Hence
P
hA, li ∈ KO .
t
u
Lemma 7. Let O1 , O2 be an arbitrary subsets of the set of objects such that
P P
O1 ⊆ O2 . Then KO 1
⊇ KO 2
.
Proof. Because of O1 ⊆ O2 is
{hA1 , l1 i ∈ P(A) × L : (∃B1 ⊇ O1 )hB1 , A1 , l1 i ∈ KP } ⊇
⊇ {hA2 , l2 i ∈ P(A) × L : (∃B2 ⊇ O2 )hB2 , A2 , l2 i ∈ KP }.
Hence
P P
KO 1
⊇ KO 2
.
t
u
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,
e the
fuzzy-subset of the set of attributes, have to be proven that O is the subset of
⇓ (A) e is the fuzzy-subset of ⇑ (O).
e if, and only if A
⇒
[
P
O ⊆⇓ (A)
e = {B ⊆ O : (∀b ∈ A)(∃hA, li ∈ KB )b ∈ A & l ≥ A(b)}.
e
Let a ∈ A be an arbitrary attribute. The lemma 14 implies that there exists
Aa ⊆ A, la ∈ L such that a ∈ Aa , la ≥ A(a)
e and
hAa , la i ∈ KP e .
⇓(A)
e implies that KP ⊇ KP
O ⊆⇓ (A) P
. Hence hAa , la i ∈ KO . So
O ⇓(A)
e
P
A(a)
e ≤ la ≤ sup{l ∈ L : (∃hA, li ∈ KO )a ∈ A} =⇑ (O)(a).
Proto-fuzzy Concepts, their Retrieval and Usage 89
Because of arbitrarity of attribute a and from unequality above implies that A
e
is the fuzzy-subset of ⇑ (O).
⇐ Let a ∈ A be an arbitrary attribute. Denote
P
la =⇑ (O)(a) = sup{l ∈ L : (∃hA, li ∈ KO )a ∈ A}.
The proposition implies that for every a ∈ A, A(a)
e ≤ la . The lemma 15 implies
P
that there exists Aa ⊆ A such that hAa , la i ∈ KO , and that implies
P
O ∈ {B ⊆ O : (∀b ∈ A)(∃hA, li ∈ KB )a ∈ A & l ≥ A(b)}
e
hence
[
P
O⊆ {B ⊆ O : (∀b ∈ A)(∃hA, li ∈ KB )a ∈ A & l ≥ A(b)}
e =⇓ (A).
e
So the set O is subset of ⇓ (A).
e t
u
For the case of object fuzzy side will be used mappings:
⇑: 2A → LO ,
⇓: LO → 2A .
e be a fuzzy subset of objects and A ⊆ A is subset of attributes.
Let O
P
⇑ (A)(o) = sup{l ∈ L : (∃hO, li ∈ KA )o ∈ O}
[
⇓ (O)
e = {T ⊆ A : (∀o ∈ O)(∃hO, li ∈ KTP )o ∈ O & l ≥ O(a)},
e
where
P
KA = {hO, li : (∃T ⊇ A)hO, T, li ∈ KP }.
Example 1. For example take the fuzzy-subset of the set of attributes,
e = {(Ma,1), (Sl,3), (Ph,1), (Ge,3), (Bi,4), (Gr,2), (En,2), (Ch,2), (Ae,4), (Hi,4)}.
A
In the table below are some proto-fuzzy concepts which contains students whose
results satisfy to A. e ={F,L,M,E}. Elements of K P
e Hence ⇓ (A) are shown in
⇓(A)
e
the next table. Hence
⇑ (⇓ (A))
e =
= {(Ma,1),(Sj,3),(Ph,1),(Ge,3),(Bi,2),(Gr,2),(En,2),(Ch,2),(Ae,2),(Hi,2)}
.
90 Ondrej Krı́dlo, Stanislav Krajči
Table 3. Some of proto-fuzzy concepts which satisfy to A
e
{M, E} {Ma,Sl,Ph,Ge,Bi,Gr,En,Ch,Ae,Hi} 1
{M, E, F} {Ma, Sl, Ph, Gr, Ae, Hi} 1
{M, E, L} {Ma, Ph, Ch} 1
{M, E, F, L} {Ma, Ph} 1
{M, E, F} {Ma,Sl,Ph,Bi,Ch,Ae,En,Gr,Hi} 2
{M, E, L} {Ma,Ph,Ge,Bi,Ch,Ae,En,Gr,Hi} 2
{M, E, F, L} {Ma,Ph,Bi,Ch,Ae,En,Gr,Hi} 2
Table 4. Elements of the K P P
⇓(A)e = K{M,E,F,L}
{Ma,Sl,Ph,Ge,Bi,Gr,En,Ch,Ae,Hi} 3
{Ma, Ch, Ae, Gr, En, Hi} 2
{Ma, Ph, Bi, Ch, Ae, Gr, En, Hi} 2
{Ma,Ph,Bi,Gr,En,Ae,Hi} 2
{Ae,Gr,En,Hi} 2
{Ph,Bi,Gr,En,Ae,Hi} 2
{Bi,Gr,En,Ae,Hi} 2
{Bi,Gr,En,Ae,Hi} 2
{Ch,Gr,En,Hi} 2
{Ma,Gr,En,Ae,Hi} 2
{Ma,Ph} 1
Proto-fuzzy Concepts, their Retrieval and Usage 91
4 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
RA = {(o, l) ∈ O × L : (∀a ∈ A)r(o, a) ≥ l}
and
RO = {(a, l) ∈ A × L : (∀o ∈ O)r(o, a) ≥ l}.
The formal context hO, L, RA i will be called object–value sight and the formal
context hA, L, RO i will be called attribute–value sight.
Table 5. Object–value and attribute–value sight
12345 12345
Fred • • • Math ••
Joey • • • Slovak language •••
Alice • • • Physics ••
Nancy • • Geography •••
Mary • • • • • Biology ••
Eve • • • • • German language • • • •
Lucy • • • English language • • • •
David • • Chemistry •••
Peter • • • Aesthetics •••
Tom • • • History ••••
Definition 9. Define new mappings
↑A : 2O → L and ↓A : L → 2O ,
↑O : 2A → L and ↓O : L → 2A .
For every O ⊆ O, A ⊆ A and l ∈ L put
↑A (O) = inf{sup{l ∈ L : (o, l) ∈ RA } : o ∈ O}
↓A (l) = {o ∈ O : (o, l) ∈ RA }
↑O (A) = inf{sup{l ∈ L : (a, l) ∈ RO } : a ∈ A}
↓O (l) = {a ∈ A : (a, l) ∈ RO }.
92 Ondrej Krı́dlo, Stanislav Krajči
Theorem 4. Pairs of mappings (↑A , ↓A ) and (↑O , ↓O ) form Galois connections
between the power-set lattice P(O) or P(A) and the lattice of values L.
Proof. The proof will be shown only for first pair. The proof for second pair is
likewise.
1. Let O1 ⊆ O2 ⊆ O. It follows from an inclusion above that
{sup{l ∈ L : (o, l) ∈ RA } : o ∈ O1 } ⊆
⊆ {sup{l ∈ L : (o, l) ∈ RA } : o ∈ O2 }
and from a properties of infimum
inf{sup{l ∈ L : (o, l) ∈ RA } : o ∈ O1 } ≥
≥ inf{sup{l ∈ L : (o, l) ∈ RA } : o ∈ O2 }.
Hence
↑A (O1 ) ≥↑A (O2 ).
2. Let l1 , l2 ∈ L. If l1 ≤ l2 then
{o ∈ O : (o, l1 ) ∈ RA } ⊇ {o ∈ O : (o, l2 ) ∈ RA }.
Hence
↓A (l1 ) ⊇↓A (l2 ).
3. Let O ⊆ O. Denote
so = sup{l ∈ L : (o, l) ∈ RA },
for arbitrary object o ∈ O. From definition of ↑A
↑A (O) = inf{sb : b ∈ O} ≤ so
and from property 2 implies
↓A (↑A (O)) ⊇↓A (so ) = {b ∈ O : (b, so ) ∈ RA }.
Arbitrarity of o implies that
[
↓A (↑A (O)) ⊇ {b ∈ O : (b, so ) ∈ RA } ⊇ O.
o∈O
4. Let l ∈ L be an arbitrary value. Denote so = sup{k ∈ L : (o, k) ∈ RA }. For
all
o ∈↓A (l) = {b ∈ O : (b, l) ∈ RA }
is so ≥ l. Hence
↑A (↓A (l)) = inf{sb : b ∈↓A (l)} ≥ l.
t
u
Proto-fuzzy Concepts, their Retrieval and Usage 93
Definition 10. The pair hO, li is called A-concept of the object–value sight
hO, L, RA i iff ↑A (O) = l and ↓A (l) = O. The set of all A-concepts will be
denoted KA .
Definition 11. The pair hA, li is called O-concept of the attribute–value sight
hA, L, RO i iff ↑O (A) = l and ↓O (l) = A. The set of all O-concepts will be
denoted KO .
It can be defined an object–value sight for every subset of attributes or
attribute–value sight for every subset of objects, but their usage for this pa-
per wasn’t necessary.
Theorem 5. Let l ∈ L, A1 , A2 ⊆ A, O1 , O2 ⊆ O such that hO, A1 , li, hO1 , A, li
∈ KP and hO2 , A2 i ∈ Kl for context hO \ O1 , A \ A1 , Rl i. Then
hO1 ∪ O2 , A1 ∪ A2 , li, ∈ KP .
Proof. It will be shown that A1 ∪ A2 =↓l (O1 ∪ O2 ) and O1 ∪ O2 =↑l (A1 ∪ A2 ).
If a ∈ A1 then for all o ∈ O is (o, a) ∈ Rl .
If a ∈ A2 then for all o ∈ O1 ∪ O2 is (o, a) ∈ Rl .
If a ∈ A1 ∪ A2 then for all o ∈ (O ∩ (O1 ∪ O2 )) = O1 ∪ O2 is (o, a) ∈ Rl .
Hence A1 ∪ A2 ⊆↑l (O1 ∪ O2 ).
The opposite inclusion will be shown by contradiction. Let us assume
a ∈↑l (O1 ∪ O2 ) and a 6∈ A1 ∪ A2 .
From a ∈↑l (O1 ∪ O2 ) implies that for all o ∈ O1 ∪ O2 ⊇ O2 is (o, a) ∈ Rl .
From a 6∈ A1 ∪ A1 implies that a ∈ (A \ (A1 ∪ A1 )) = ((A \ A1 ) \ A2 ). It is the
contradiction to precondition hO2 , A2 i ∈ Kl for context hO \ O1 , A \ A1 , Rl i.
The second equality can be shown likewise. t
u
Subcontexts from the theorem will be called auxiliary subcontexts of l-cut.
Concepts of sights will be retrieved with a help of mappings ↑A , ↓A , ↑O and ↓O .
It’s good to know that hO, li ∈ KA then hO, A, li ∈ KP , because of A is closed.
Denote A as the set of all subjects and O as the group of all students from our
example. Hence
hO, A, 4i ∈ KP ,
hO \ {N,D}, A, 3i, hO, A \ {Ma,Ph,Bi}, 3i ∈ KP ,
h{M,E}, A, 1i, hO, {Gr,En,Hi}, 2i ∈ KP ,
Let us create auxiliary subcontexts of 3-cut, 2-cut and 1-cut.
Table 6. Auxiliary subcontexts of 3-cut
3 Ma Ph Bi
N •
D •
94 Ondrej Krı́dlo, Stanislav Krajči
There are only two concepts in the auxiliary subcontext of 3-cut, h{N }, {Bi}i
and h{D}, {M a}i. The theorem 23 implies that
hO \ {N}, A \ {P h, Bi}, 3i, hO \ {D}, A \ {Ma,Ph}, 3i ∈ KP .
Table 7. Auxiliary subcontext of 2-cut
2 Ma Sl Ph Ge Bi Ch Ae
F • • • • • •
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 h{M,E}, A, 1i, hO, {Gr,En,Hi}, 2i ∈
KP .
Table 8. Auxiliary subcontext of 1-cut
1 Ma Sl Ph Ge Bi Gr En Ch Ae Hi
F • • • • •
J • • • • • • •
A • • • •
N •
L • • •
D • •
P • • • •
T • •
5 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.
Proto-fuzzy Concepts, their Retrieval and Usage 95
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