=Paper=
{{Paper
|id=None
|storemode=property
|title=CRL-Chu Correspondences
|pdfUrl=https://ceur-ws.org/Vol-1062/paper9.pdf
|volume=Vol-1062
|dblpUrl=https://dblp.org/rec/conf/cla/KridloO13
}}
==CRL-Chu Correspondences==
https://ceur-ws.org/Vol-1062/paper9.pdf
CRL-Chu correspondences
Ondrej Krı́dlo1 and Manuel Ojeda-Aciego2
1
University of Pavol Jozef Šafárik, Košice, Slovakia?
2
Dept. Matemática Aplicada, Univ. Málaga, Spain??
Abstract. We continue our study of the general notion of L-Chu cor-
respondence by introducing the category CRL-ChuCors incorporating
residuation to the underlying complete lattice L, specifically, on the ba-
sis of a residuation-preserving isotone Galois connection λ. Then, the
L-bonds are generalized within this same framework, and its structure
is related to that of the extent of a suitably defined λ-direct product.
1 Introduction
Morphisms have been suggested [7] as fundamental structural properties for the
modelling of, among other applications, communication, data translation, and
distributed computing. Our approach can be seen within a research topic linking
concept lattices with the theory of Chu spaces [10, 11]; in the latter, it is shown
that the notion of state in Scott’s information system corresponds precisely to
that of formal concepts in FCA with respect to all finite Chu spaces, and the
entailment relation corresponds to association rules (another link between FCA
with database theory) and, specifically, on the identification of the categories
associated to certain constructions.
Other researchers have studied as well the relationships between Chu con-
structions and L-fuzzy FCA. For instance, in [1] FCA is linked to both order-
theoretic developments in the theory of Galois connections and to Chu spaces;
as a result, not surprisingly from our previous works, they obtain further rela-
tionships between formal contexts and topological systems within the category
of Chu systems. Recently, Solovyov, in [9], extends the results of [1] to clar-
ify the relationships between Chu spaces, many-valued formal contexts of FCA,
lattice-valued interchange systems and Galois connections.
This work is based on the notion, introduced by Mori in [8], of Chu corre-
spondences as morphisms between formal contexts. This categorical approach
has been used in previous works [3, 5, 6]. For instance, in [6], the categories asso-
ciated to L-formal contexts and L-CLLOS were defined and a constructive proof
was given of the equivalence between the categories of L-formal contexts with L-
Chu correspondences as morphisms and that of completely lattice L-ordered sets
and their corresponding morphisms. Similar results can be found in [2], where a
?
Partially supported by grant VEGA 1/0832/12 and APVV-0035-10.
??
Partially supported by Spanish Ministry of Science and FEDER funds through
project TIN09-14562-C05-01 and Junta de Andalucı̈¿ 21 ı̈¿ 12 a project P09-FQM-5233.
c paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA
2013, pp. 105–116, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La
Rochelle, 2013. Copying permitted only for private and academic purposes.
106 Ondrej Krı́dlo and Manuel Ojeda-Aciego
new notion of morphism on formal contexts resulted in a category equivalent to
both the category of complete algebraic lattices and Scott continuous functions,
and a category of information systems and approximable mappings.
We are concerned with the category of fuzzy formal contexts and λ-Chu
correspondences, built on the basis of a residuation-preserving isotone Galois
connection λ. Then, the corresponding extension of the notion of bond between
contexts is generalized to this framework, and its properties are studied.
2 Preliminaries
2.1 Residuated lattice
Definition 1. A complete residuated lattice is an algebra hL, ∧, ∨, 0, 1, ⊗, →i
where
– hL, ∧, ∨, 0, 1i is a complete lattice with the top 1 and the bottom 0,
– hL, ⊗, 1i is a commutative monoid,
– h⊗, →i is an adjoint pair, i.e. for any a, b, c ∈ L:
a ⊗ b ≤ c is equivalent to a ≤ b → c
Definition 2. A complete residuated lattice L = hL, ∧, ∨, 0, 1, ⊗, →i such that
for any value k ∈ L holds ¬¬k = k where ¬k = k → 0 is said to be endowed
with double negation law.
Lemma 1. Let L be a complete residuated lattice satisfying the double negation
law. Then for any k, m ∈ L holds ¬k → m = ¬m → k.
2.2 Basics of Fuzzy FCA
Definition 3. An L-fuzzy formal context C is a triple hB, A, L, ri, where B, A
are sets, L is a complete residuated lattice, and r : B × A → L is an L-fuzzy
binary relation.
Definition 4. Let C = hB, A, L, ri be an L-fuzzy formal context. A pair of
derivation operators h↑, ↓i of the form ↑: LB → LA and ↓: LA → LB , is de-
fined as follows
^
↑ (f )(a) = (f (b) → r(b, a)) for any f ∈ LB and a ∈ A,
b∈B
^
↓ (g)(b) = (g(a) → r(b, a)) for any g ∈ LA and b ∈ B.
a∈A
Lemma 2. Let h↑, ↓i be a pair of derivation operators defined on an L-fuzzy for-
mal context C. A pair h↑, ↓i forms a Galois connection between complete lattices
of all L-sets of objects LB and attributes LA .
CRL-Chu correspondences 107
Definition 5. Let C = hB, A, L, ri be an L-fuzzy formal context. A formal con-
cept is a pair of L-sets hf, gi ∈ LB × LA such that ↑ (f ) = g and ↓ (g) = f .
The set of all L-concepts of C will be denoted as FCL(C). The object (resp. at-
tribute) part of any concept is called extent (resp. intent). The sets of all extents
or intents of C will be denoted as Ext(C) or Int(C), respectively.
2.3 L-Bonds and L-Chu correspondences
Definition 6. Let X and Y be two sets. An L-multifunction from X to Y is
said to be a mapping from X to LY .
Definition 7. Let Ci = hBi , Ai , L, ri i for i ∈ {1, 2} be two L-fuzzy formal con-
texts. A pair of L-multifunctions ϕ = hϕL , ϕR i such that
– ϕL : B1 −→ Ext(C2 ),
– ϕR : A2 −→ Int(C1 ),
where ↑2 (ϕL (o1 ))(a2 ) = ↓1 (ϕR (a2 ))(o1 ) for any (o1 , a2 ) ∈ B1 × A2 , is said to be
an L-Chu correspondence between C1 and C2 . A set of all L-Chu corresepondences
between C1 and C2 will be denoted by L-ChuCors(C1 , C2 ).
Definition 8. Let Ci = hBi , Ai , L, ri i for i ∈ {1, 2} be two L-fuzzy formal con-
texts. An L-multifunction β : B1 −→ Int(C2 ), such that β t : A2 −→ Ext(C1 ),
where β t (a2 )(o1 ) = β(o1 )(a2 ) for any (o1 , a2 ) ∈ B1 × A2 , is said to be an L-
bond. A set of all L-bonds between C1 and C2 will be denoted by L-Bonds(C1 , C2 ).
Lemma 3. Let Ci = hBi , Ai , L, ri i for i ∈ {1, 2} be two L-fuzzy formal con-
texts. The sets L-Bonds(C1 , C2 ) and L-ChuCors(C1 , C2 ) form complete lattices
and, moreover, there exists a dual isomorphism between them.
3 Residuation-preserving isotone Galois connections
Definition 9. An isotone Galois connection between two complete lattices L1 =
(L1 , ≤1 ) and L2 = (L2 , ≤2 ) is a pair of monotone mappings λ = hλL , λR i with
λL : L1 −→ L2 and λR : L2 −→ L1
such that, for any k1 ∈ L1 and k2 ∈ L2 , the following equivalence holds
k1 ≤1 λR (k2 ) ⇐⇒ λL (k1 ) ≤2 k2 . (1)
The general theory of adjunctions provides the following result:
Lemma 4. Let hλL , λR i be an isotone Galois connection, then for all k1 ∈ L1
and k2 ∈ L2
_
λR (k2 ) = {m ∈ L1 : λL (m) ≤2 k2 } (2)
^
λL (k1 ) = {m ∈ L2 : k1 ≤1 λR (m)} (3)
108 Ondrej Krı́dlo and Manuel Ojeda-Aciego
Definition 10. An isotone Galois connection λ between two complete residuated
lattices L1 = (L1 , ⊗1 , →1 ) and L2 = (L2 , ⊗2 , →2 ) is said to be a residuation-
preserving isotone Galois connection if for any k1 , m1 ∈ L1 and k2 , m2 ∈ L2 the
following equalities hold:
λL (k1 ⊗1 m1 ) = λL (k1 ) ⊗2 λL (m1 ) (4)
λR (k2 ⊗2 m2 ) = λR (k2 ) ⊗1 λR (m2 ) (5)
k2 →2 λL (m1 ) ≥2 λL (λR (k2 ) →1 m1 ) (6)
The set of all residuation-preserving isotone Galois connections from L1 to L2
will be denoted as CRL(L1 , L2 ).
There is no need to consider other →-preserving rules, since they follow from
the previous ones, as stated by the following lemmas.
Lemma 5. For all k ∈ L1 and m ∈ L2 the following equality holds
k →1 λR (m) = λR (λL (k) →2 m) (7)
Proof. Consider the following chain of equivalences
(1)
l ⊗1 k ≤1 λR (m) ⇐⇒ λL (l ⊗1 k) ≤2 m
(4)
⇐⇒ λL (l) ⊗2 λL (k) ≤2 m
(adj)
⇐⇒ λL (l) ≤2 λL (k) →2 m
As a result, we can write
_
k →1 λR (m) = {l ∈ L1 : l ⊗1 k ≤ λR (m)}
_
= {l ∈ L1 : λL (l) ≤ λL (k) →2 m}
(2)
= λR (λL (k) →2 m)
It is worth to note that this proof does not work in the case of (6) because, for
the construction of λL , one had to use (3) instead of (2). t
u
Lemma 6. For all ki , mi ∈ Li for i ∈ {1, 2}, the following inequalities hold
λL (k1 →1 m1 ) ≤2 λL (k1 ) →2 λL (m1 ) (8)
λR (k2 →2 m2 ) ≤1 λR (k2 ) →1 λR (m2 ) (9)
Proof. By the adjoint property and the following chain of inequalities
(4)
λL (k1 →1 m1 ) ⊗2 λL (k1 ) = λL ((k1 →1 m1 ) ⊗1 k1 ) ≤2 λL (m1 )
Similarly, we obtain the other one. t
u
CRL-Chu correspondences 109
Below, we recall the notion of fixpoint of a Galois connection, the definition
is uniform to the different types of Galois connection, either antitone or isotone,
or with any other extra requirement.
Definition 11. Let λ be a Galois connection between complete residuated lat-
tices L1 and L2 . The set of all fixpoints of λ is defined as
FPλ = {hk1 , k2 i ∈ L1 × L2 : λL (k1 ) = k2 , λR (k2 ) = k1 }.
Lemma 7. Given λ ∈ CRL(L1 , L2 ), the set of its fixpoints can be provided with
the structure of complete residuated lattice Φλ = hFPλ , ∧, ∨, 0, 1, ⊗, →i where
0 = hλR (02 ), 02 i, 1 = h11 , λL (11 )i, and ⊗ and → are defined componentwise.
Proof. We have to check just that the componentwise operations provide a resid-
uated structure to the set of fixed point of λ.
Conditions (4) and (5) allow to prove that componentwise product ⊗ is a
closed operation in FPλ , whereas condition (6) allows to prove that the compo-
nentwise implication is a closed operation in FPλ .
It is not difficult to show that, in fact, hFPλ , ⊗, 1i is a commutative monoid:
commutativity and associativity follow directly; for the neutral element just
consider the following chain of equalities: For any hk1 , k2 i ∈ FPλ holds
hk1 , k2 i ⊗ h11 , λL (11 )i = hk1 ⊗1 11 , λL (k1 ) ⊗2 λL (11 )i
= hk1 , λL (k1 ⊗1 11 )i
= hk1 , λL (k1 )i = hk1 , k2 i
The adjoint property follows by definition. t
u
4 CRL-Chu correspondences and their category
In this section, the notion of L-Chu correspondence is generalized on the basis
of a residuation-preserving isotone Galois connection λ. The formal definition is
the following:
Definition 12. Let Ci = hBi , Ai , Li , ri i for i ∈ {1, 2} be two fuzzy formal
contexts, and consider λ ∈ CRL(L1 , L2 ). A pair of fuzzy multifunctions ϕ =
hϕL , ϕR i of types
ϕL : B1 −→ Ext(C2 ) and ϕR : A2 −→ Int(C1 )
such that for any (o1 , a2 ) ∈ B1 × A2 the following inequality holds
λL (↓1 (ϕR (a2 ))(o1 )) ≤2 ↑2 (ϕL (o1 ))(a2 ) (10)
is said to be a λ-Chu correspondence.
Note that (10) is equivalent to ↓1 (ϕR (a2 ))(o1 ) ≤1 λR (↑2 (ϕL (o1 ))(a2 )).
110 Ondrej Krı́dlo and Manuel Ojeda-Aciego
It is not difficult to check that the definition of λ-Chu correspondence generalizes
the previous one based on a complete (residuated) lattice L; formally, we have
the following
Definition 13. Let X be an arbitrary set. Mapping idX defined by idX (x) = x
for any x ∈ X is said to be an identity mapping on X.
Lemma 8. Any L-Chu correspondence is a hidL , idL i-Chu correspondence.
We are now in position to define the category of parameterized fuzzy formal
contexts and λ-Chu correspondences between them:
Definition 14. We introduce a new category whose objects are parameterized
fuzzy formal contexts hB, A, L, ri and λ-Chu correspondences between them.
The identity arrow of an object hB, A, L, ri is the hidL , idL i-Chu correspon-
dence ι such that
– ιL (o) = ↓↑ (χo ) for any o ∈ B
– ιR (a) = ↑↓ (χa ) for any a ∈ A
– where χx (x) = 1 and χx (y) = 0 for any y 6= x.
Composition of arrows3 hλ, ϕi : C1 → C2 and hµ, ψi : C2 → C3 where Ci =
hBi , Ai , Li , ri i for i ∈ {1, 2, 3} is defined as:
– (hµ, ψi ◦ hλ, ϕi)L (o1 ) = ↓3 ↑3 (ψL+ (ϕL (o1 )))
– (hµ, ψi ◦ hλ, ϕi)R (a3 ) = ↑1 ↓1 (ϕR+ (ψR (a3 )))
where, for any (oi , ai ) ∈ Bi × Ai , i ∈ {1, 3},
_
ψL+ (ϕL (o1 ))(o3 ) = ψ(o2 )(o3 ) ⊗3 µL (ϕL (o1 )(o2 ))
o2 ∈B2
_
ϕR+ (ψR (a3 ))(a1 ) = ϕR (a2 )(a1 ) ⊗1 λR (ψR (a3 )(a2 ))
a2 ∈A2
Obviously, one has to check that the proposed notions of composition and iden-
tity are well-defined, and this is stated in the following lemmas.
Lemma 9. The identity arrow of any fuzzy formal context hB, A, L, ri is a
hidL , idL i-Chu correspondence.
Lemma 10. Consider hλ, ϕi : C1 → C2 and hµ, ψi : C2 → C3 , then hµ, ψi ◦ hλ, ϕi
is a (µ ◦ λ)-Chu correspondence. Moreover, composition of λ-correspondences is
associative.
3
Any λ-Chu correspondence ϕ can be conveniently denoted by hλ, ϕi.
CRL-Chu correspondences 111
5 λ-Bonds and λ-direct product of two contexts
We proceed with the corresponding extension of the notion of bond between
contexts, and the study of its properties.
Definition 15. Given a multifunction ω : X → (L1 × L2 )Y , its projections ω i
for i ∈ {1, 2} are defined by ω i (x)(y) = ki , provided that ω(x)(y) = (k1 , k2 ).
Transposition of such multifunction is defined by ω t (y)(x) = ω(x)(y).
Definition 16. Given two fuzzy formal contexts Ci = hBi , Ai , Li , ri i, i ∈ {1, 2},
and λ ∈ CRL(L1 , L2 ). A λ-bond is a multifunction β : B1 → (L1 × L2 )A2 such
that, for any (o1 , a2 ) ∈ B1 × A2 :
β 2 (o1 ) is an intent of C2 (11)
t 1
(β ) (a2 ) is an extent of C1 (12)
1 2 1 2
β (o1 )(a2 ) ≤1 λR (β (o1 )(a2 )) or equivalently λL (β (o1 )(a2 )) ≤2 β (o1 )(a2 )
(13)
The known relation between L-bonds and L-Chu correspondences is pre-
served in the λ-case. Formally,
Lemma 11. Let β be a λ-bond between two fuzzy contexts Ci = hBi , Ai , Li , ri i
for i ∈ {1, 2}. Then ϕβ defined as
ϕβL (o1 ) = ↓2 (β 2 (o1 )) (14)
t 1
ϕβR (a2 ) = ↑1 ((β ) (a2 )) (15)
is λ-Chu correspondence.
Proof. By calculation
(14) (11)
↑2 (ϕβL (o1 ))(a2 ) = ↑2 ↓2 (β 2 (o1 ))(a2 ) = β 2 (o1 )(a2 )
(13)
≥ λL (β 1 (o1 )(a2 )) = λL ((β t )1 (a2 )(o1 ))
(12)
= λL (↓1 ↑1 ((β t )1 (a2 ))(o1 ))
(15)
= λL (↓1 (ϕβR (a2 ))(o1 ))
t
u
Lemma 12. Let L1 , L2 be two complete residuated lattices satisfying the double
negation law and let λ ∈ CRL(L1 , L2 ). Then Φλ satisfies double negation law.
Proof. Consider an arbitrary hk1 , k2 i ∈ FPλ . We have that, by definition,
¬¬hk1 , k2 i = (hk1 , k2 i → 0) → 0 = (hk1 , k2 i → hλR (02 ), 02 i) → hλR (02 ), 02 i
= h(k1 →1 λR (02 )) →1 λR (02 ), (k2 →2 02 ) →2 02 i
112 Ondrej Krı́dlo and Manuel Ojeda-Aciego
The result for the second component is obvious; for the first one, taking into
account that (λR (02 ), 02 ) is a fixed point, we have
λL (k1 ) = k2 = (k2 →2 02 ) →2 02
= k2 →2 λL λR (02 ) →2 λL λR (02 )
(6)
≥ 2 λL (λR λL (λR (k2 ) →1 λR (02 )) →1 λR (02 ))
(?)
= λL ((λR (k2 ) →1 λR (02 )) →1 λR (02 ))
= λL ((k1 →1 λR (02 )) →1 λR (02 ))
(∗)
≥ 2 λL (k1 )
where equality (?) follows because, by Lemma 7, λR (k2 ) →1 λR (02 ) is a closed
value in L1 for the composition λR λL , and inequality (∗) follows from the mono-
tonicity of λL .
As a result of the previous chain we obtain the following equality
λL (k1 ) = λL ((k1 →1 λR (02 )) →1 λR (02 ))
and, again by Lemma 7, since k1 and λR (02 ) are closed for λR λL , as a result
(k1 →1 λR (02 )) →1 λR (02 ) is closed too, and k1 = (k1 →1 λR (02 )) →1 λR (02 ).
t
u
We are now ready to include the characterization result on the structure of
λ-bonds, but we have to introduce the notion of λ-direct product of contexts.
Definition 17. Let Ci = hBi , Ai , Li , ri i for i ∈ {1, 2} be two fuzzy formal
contexts, λ ∈ CRL(L1 , L2 ) and L1 , L2 satisfy the double negation law. The
fuzzy formal context hB1 × A2 , B2 × A1 , Φλ , ∆λ i where ∆λ ((o1 , a2 ), (o2 , a1 )) =
¬(λ1 (r1 )) → λ2 (r2 (o2 , a2 )) is said to be the λ-direct product of C1 and C2 , where
λ1 (k) = hλR λL (k), λL (k)i for all k ∈ L1 (16)
λ2 (k) = hλR (k), λL λR (k)i for all k ∈ L2 (17)
Lemma 13. Let C1 ∆λ C2 be the λ-direct product of fuzzy formal contexts C1
and C2 , and λ ∈ CRL(L1 , L2 ). For any extent of C1 ∆λ C2 there exists a λ-bond
between C1 and C2 .
1 ×A2
Proof. Let hβ, γi be a concept of C1 ∆λ C2 . Then β ∈ FPB
λ ⊆ (L1 ×L2 )B1 ×A2 .
β(o1 , a2 ) = ↓∆λ (γ)(o1 , a2 )
^ ^
= (γ(o2 , a1 ) → ∆λ ((o1 , a2 ), (o2 , a1 )))
o2 ∈B2 a1 ∈A1
^ ^
= (γ(o2 , a1 ) → (¬λ1 (r1 (o1 , a1 ))) → λ2 (r2 (o2 , a2 )))
o2 ∈B2 a1 ∈A1
CRL-Chu correspondences 113
Then
^ ^
β 1 (o1 , a2 ) = (γ 1 (o2 , a1 ) →1 (¬λR λL (r1 (o1 , a1 ))) →1 λR (r2 (o2 , a2 )))
o2 ∈B2 a1 ∈A1
^ ^
= ((γ 1 (o2 , a1 ) ⊗1 ¬λR λL (r1 (o1 , a1 ))) →1 λR (r2 (o2 , a2 )))
o2 ∈B2 a1 ∈A1
(7) ^ ^
= λR (λL (γ 1 (o2 , a1 ) ⊗1 ¬λR λL (r1 (o1 , a1 ))) →2 r2 (o2 , a2 ))
o2 ∈B2 a1 ∈A1
^ ^
= λR ( (λL (γ 1 (o2 , a1 ) ⊗1 ¬λR λL (r1 (o1 , a1 ))) →2 r2 (o2 , a2 )))
o2 ∈B2 a1 ∈A1
^ _
= λR ( ( (λL (γ 1 (o2 , a1 ) ⊗1 ¬λR λL (r1 (o1 , a1 )))) →2 r2 (o2 , a2 )))
o2 ∈B2 a1 ∈A1
^
= λR ( (σ(o1 )(o2 ) →2 r2 (o2 , a2 )))
o2 ∈B2
= λR (↑2 (σ(o1 ))(a2 ))
Similarly
^ ^
β 2 (o1 , a2 ) = ((γ 2 (o2 , a1 ) ⊗2 ¬λL λR (r2 (o2 , a2 ))) →2 λL (r1 (o1 , a1 )))
o2 ∈B2 a1 ∈A1
(6) ^ ^
≥2 λL (λR (γ 2 (o2 , a1 ) ⊗2 ¬λL λR (r2 (o2 , a2 ))) →2 r1 (o1 , a1 ))
o2 ∈B2 a1 ∈A1
^ ^
≥2 λ L ( (λR (γ 2 (o2 , a1 ) ⊗2 ¬λL λR (r2 (o2 , a2 ))) →1 r1 (o1 , a1 )))
o2 ∈B2 a1 ∈A1
^ _
= λL ( ( (λR (γ 2 (o2 , a1 ) ⊗2 ¬λL λR (r2 (o2 , a2 )))) →1 r1 (o1 , a1 )))
a1 ∈A1 o2 ∈B2
^
= λL ( (τ (a2 )(a1 ) →1 r1 (o1 , a1 ))) = λL (↓1 (τ (a2 ))(o1 ))
a1 ∈A1
Then let us define a multifunction βb : B1 −→ (L1 × L2 )A2 as follows
βb2 (o1 ) = ↑2 (σ(o1 ))
(βbt )1 (a2 ) = ↓1 (τ (a2 ))
where σ and τ are multifunctions above. We see that βb2 (o1 ) is the intent of C2 ,
(βbt )1 (a2 ) is the extent of C1 and moreover β(o1 , a2 ) ∈ FPλ , hence λL (β 1 (o1 , a2 )) =
β 2 (o1 , a2 ) and
λL ((βbt )1 (a2 )(o1 )) = λL (↓1 (τ (a2 )))
≤2 β 2 (o1 , a2 )
= λL (β 1 (o1 , a2 ))
= λL λR (↑2 (σ(o1 ))(a2 ))
≤2 ↑2 (σ(o1 ))(a2 ) = βb2 (o1 )(a2 )
114 Ondrej Krı́dlo and Manuel Ojeda-Aciego
Therefore βb is a λ-bond between C1 and C2 . t
u
6 Motivation example
Lets have two tables of the following data. First table of students, school subjects
and study results. Second table of universities (or areas of study) and their
requirements for results of students. We would like to find the assignment of
students and universities that depends on study results and requirements of
universities.
U CS Tech Med
S Math Phi Chem Bio
Math Ex G W
Anna A A B C
Phi G Ex G
Boris C B A B
Chem W Ex Ex
Cyril D E B C
Bio W W Ex
First table is filled by degrees from well known structure {A,B,C,D,E,F} where
A is best and F means failed. Second one is filled by degrees {Ex,G,W} that
means Ex-excelent, G-good, W-weak. Now lets define a λ-translation between
such truth-degrees structures.
A B C D E F Ex G W
λL (−) Ex G W W W W λR (−) A B C
hλL , λR i is an isotone Galois connection. In fact, hλL , λR i is a residuation-
preserving isotone Galois connection over Lukasiewicz logic, whose set of fix-
points is
FPλ = {(A, Ex); (B, G); (C, W)}
The λ-direct product S∆λ U is the following table that has 510 concepts. Lets
simplify the table with translation (A,Ex) as 1, (B,G) as 0.5 and (C,W) as 0.
1 1 0.5 0 1 1 1 0.5 1 1 1 1 1 1 1 1
1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 0.5 0
1 1 1 1 1 1 1 0.5 1 1 0.5 0 1 1 0.5 0
0 0.5 1 0.5 0.5 1 1 1 1 1 1 1 1 1 1 1
0.5 1 1 1 1 1 1 1 1 1 1 1 0 0.5 1 0.5
1 1 1 1 0.5 1 1 1 0 0.5 1 0.5 0 0.5 1 0.5
0 0 0.5 0 0.5 0.5 1 0.5 1 1 1 1 1 1 1 1
0.5 0.5 1 0.5 1 1 1 1 1 1 1 1 0 0 0.5 0
1 1 1 1 0.5 0.5 1 0.5 0 0 0.5 0 0 0 0.5 0
Extents of S∆λ U are tables of the form students×universities and intents are
tables of the form subjects×subjects. One of the concepts that their intents has
1 on diagonal (it means that any subject is assigned to itself) is shown below.
CRL-Chu correspondences 115
Math Phi Chem Bio
Med Tech CS
Math A,Ex A,Ex B,G C,W
Anna B,G B,G A,Ex
Phi B,G A,Ex A,Ex B,G
Boris A,Ex B,G C,W
Chem C,W B,G A,Ex B,G
Cyril B,G B,G C,W
Bio C,W B,G A,Ex B,G
Such concept should be translated into {1; 0.5; 0} structure.
Math Phi Chem Bio
Med Tech CS
Math 1 1 0.5 0
Anna 0.5 0.5 1
Phi 0.5 1 1 0.5
Boris 1 0.5 0
Chem 0 0.5 1 0.5
Cyril 0.5 0.5 0
Bio 0 0.5 1 0.5
Now we can see the result. Due to results of students and requirements of
universities we can advise Anna to study Computer science; similarly, we can
advise Boris to study Medicine or Technical area; finally, it is hard to advise
anything to Cyril. The assignment is right as it is obvious from study results.
Relaxing the connection
Let’s change the λ-connection between such truth degrees structures as follows:
A BC D E F Ex G W
λL (−) Ex G G W W W λR (−) A B D
The set of fixpoints is FPλ = {(A, Ex); (B, G); (D, W)} such that is easy to
translate (A, Ex) as 1, (B, G) as 0.5 and (D, W) as 0. The direct product is
shown below, and has 104 concepts.
1 1 1 1 1 1 1 1 1 1 0.5 0.5 1 1 0.5 0.5
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.5 0.5
1 1 0.5 0.5 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0.5 0.5 1 0.5 0.5 0.5 1 0.5
1 1 1 1 1 1 1 1 1 1 1 1 0.5 0.5 1 0.5
0.5 0.5 1 0.5 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0.5 0.5 1 1 0 0 0.5 0.5 0 0 0.5 0.5
0.5 0.5 1 1 1 1 1 1 1 1 1 1 0 0 0.5 0.5
0 0 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 1
We have chosen one with 1-diagonal in the intent.
Math Phi Chem Bio
Med Tech CS
Math A,Ex A,Ex B,G B,G
Anna B,G A,Ex A,Ex
Phi A,Ex A,Ex A,Ex A,Ex
Boris A,Ex A,Ex B,G
Chem B,G B,G A,Ex B,G
Cyril B,G B,G D,W
Bio B,G B,G B,G B,G
116 Ondrej Krı́dlo and Manuel Ojeda-Aciego
Translating the context into {1; 0.5; 0} structure, we obtain
Math Phi Chem Bio
Med Tech CS
Math 1 1 0.5 0.5
Anna 0.5 1 1
Phi 1 1 1 0.5
Boris 1 1 0.5
Chem 0.5 0.5 1 0.5
Cyril 0.5 0.5 0
Bio 0.5 0.5 0.5 0.5
It can seen that our advice is more generous but still coincide to input data.
7 Conclusion
We continue our study of the general notion of L-Chu correspondence by intro-
ducing the category CRL-ChuCors incorporating residuation to the underlying
complete lattice L, specifically, on the basis of a residuation-preserving isotone
Galois connection λ. Then, the L-bonds are generalized within this same frame-
work, and its structure is related to that of the extent of a suitably defined
λ-direct product. A first relationship between extents of λ-direct product have
been proved; it is expected to find a proof of the stronger result which states
an isomorphism between the extents of the λ-direct product and the λ-bonds
between C1 and C2 .
Potential applications are primary motivations for further future work, for
instance, to consider possible classes of formal L-contexts induced from existing
datamining notions, and study its associated categories.
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