CRL-Chu Correspondences

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Ondrej Kr´ıdlo¹and Manuel Ojeda-Aciego²

1 University of Pavol Jozef ˇSaf´arik, Koˇsice, Slovakia^?

2 Dept. Matem´atica Aplicada, Univ. M´alaga, Spain^??

Abstract. We continue our study of the general notion of L-Chu correspondence by introducing the category CRL-ChuCors incorporating residuation to the underlying complete latticeL, specifically, on the basis of a residuation-preserving isotone Galois connectionλ. Then, the L-bonds are generalized within this same framework, and its structure is related to that of the extent of a suitably definedλ-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 toassociation 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 constructions 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 relationships 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 correspondences as morphisms between formal contexts. This categorical approach has been used in previous works [3, 5, 6]. For instance, in [6], the categories associated toL-formal contexts andL-CLLOS were defined and a constructive proof was given of the equivalence between the categories ofL-formal contexts withL- Chu correspondences as morphisms and that of completely latticeL-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 Andaluc¨ı¿¹₂¨ı¿¹₂a 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.

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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 connectionλ. 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,∧,∨,0,1iis a complete lattice with the top 1 and the bottom0, – hL,⊗,1iis a commutative monoid,

– h⊗,→iis an adjoint pair, i.e. for anya, b, c∈L:

a⊗b≤c is equivalent toa≤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. LetLbe 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 contextC is a triplehB, A,L, ri, whereB,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 ↑: L^B → L^A and ↓: L^A → L^B, is defined as follows

↑(f)(a) = ^

b∈B

(f(b)→r(b, a))for anyf ∈L^B anda∈A,

↓(g)(b) = ^

a∈A

(g(a)→r(b, a))for anyg∈L^A andb∈B.

Lemma 2. Leth↑,↓ibe a pair of derivation operators defined on anL-fuzzy formal contextC. A pair h↑,↓iforms a Galois connection between complete lattices of allL-sets of objectsL^B and attributes L^A.

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Definition 5. Let C=hB, A,L, ribe an L-fuzzy formal context. Aformal concept is a pair of L-sets hf, gi ∈ L^B ×L^A such that ↑(f) = g and ↓(g) =f. The set of allL-concepts of C will be denoted asFCL(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 asExt(C) orInt(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 fromX toL^Y.

Definition 7. Let Cⁱ =hBi, Ai,L, rii fori∈ {1,2} be twoL-fuzzy formal contexts. A pair of L-multifunctionsϕ=hϕL, ϕRisuch 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 anL-Chu correspondence betweenC¹andC². A set of allL-Chu corresepondences between C¹ andC² will be denoted byL-ChuCors(C¹,C²).

Definition 8. Let Cⁱ =hBi, Ai,L, rii fori∈ {1,2} be twoL-fuzzy formal contexts. 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 allL-bonds betweenC¹andC²will be denoted byL-Bonds(C¹,C²).

Lemma 3. Let Cⁱ = hBi, Ai,L, rii for i ∈ {1,2} be two L-fuzzy formal contexts. 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 connectionbetween two complete latticesL¹= (L1,≤¹)andL²= (L2,≤²)is a pair of monotone mappingsλ=hλL, λRiwith

λL: L1−→L2 and λR:L2−→L1

such that, for any k1∈L1 andk2∈L2, the following equivalence holds

k1≤¹λR(k2) ⇐⇒ λL(k1)≤²k2. (1) The general theory of adjunctions provides the following result:

Lemma 4. Let hλL, λRibe an isotone Galois connection, then for all k1 ∈L1

andk2∈L2

λR(k2) =_

{m∈L1:λL(m)≤²k2} (2) λL(k1) =^

{m∈L2:k1≤¹λR(m)} (3)

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Definition 10. An isotone Galois connectionλbetween two complete residuated lattices L¹ = (L1,⊗¹,→¹) and L² = (L2,⊗²,→²) is said to be a residuation- preserving isotone Galois connectionif for any k1, m1∈L1 andk2, m2∈L2 the following equalities hold:

λL(k1⊗¹m1) =λL(k1)⊗²λL(m1) (4) λR(k2⊗²m2) =λR(k2)⊗¹λR(m2) (5) k2→²λL(m1)≥²λL(λR(k2)→¹m1) (6) The set of all residuation-preserving isotone Galois connections from L¹ toL² will be denoted asCRL(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 allk∈L1 andm∈L2 the following equality holds

k→¹λR(m) =λR(λL(k)→²m) (7) Proof. Consider the following chain of equivalences

l⊗¹k≤¹λR(m) ⇐⇒⁽¹⁾ λL(l⊗¹k)≤²m

⇐⇒(4) λL(l)⊗²λL(k)≤²m

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⇐⇒ λL(l)≤²λL(k)→²m As a result, we can write

k→¹λR(m) =_

{l∈L1:l⊗¹k≤λR(m)}

=_

{l∈L1:λL(l)≤λL(k)→²m}

(2)= λR(λL(k)→²m)

It is worth to note that this proof does not work in the case of (6) because, for the construction ofλL, one had to use (3) instead of (2). ut Lemma 6. For allki, mi∈Li fori∈ {1,2}, the following inequalities hold

λL(k1→¹m1)≤²λL(k1)→²λL(m1) (8) λR(k2→²m2)≤¹λR(k2)→¹λR(m2) (9) Proof. By the adjoint property and the following chain of inequalities

λL(k1→¹m1)⊗²λL(k1)⁽⁴⁾=λL((k1→¹m1)⊗¹k1)≤²λL(m1)

Similarly, we obtain the other one. ut

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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- ticesL¹ andL². The set of all fixpoints of λis defined as

FPλ={hk1, k2i ∈L1×L2:λL(k1) =k2, λR(k2) =k1}.

Lemma 7. Givenλ∈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),02i,1 =h11, λL(11)i, and⊗and →are defined componentwise.

Proof. We have to check just that the componentwise operations provide a residuated structure to the set of fixed point ofλ.

Conditions (4) and (5) allow to prove that componentwise product ⊗ is a closed operation in FPλ, whereas condition (6) allows to prove that the componentwise implication is a closed operation in FPλ.

It is not difficult to show that, in fact,hFPλ,⊗,1iis a commutative monoid:

commutativity and associativity follow directly; for the neutral element just consider the following chain of equalities: For anyhk1, k2i ∈FPλ holds

hk1, k2i ⊗ h11, λL(11)i=hk1⊗¹11, λL(k1)⊗²λL(11)i

=hk1, λL(k1⊗¹11)i

=hk1, λL(k1)i=hk1, k2i

The adjoint property follows by definition. ut

4 CRL-Chu correspondences and their category

In this section, the notion ofL-Chu correspondence is generalized on the basis of a residuation-preserving isotone Galois connectionλ. The formal definition is the following:

Definition 12. Let Cⁱ = hBi, Ai,Lⁱ, rii for i ∈ {1,2} be two fuzzy formal contexts, and consider λ ∈ CRL(L¹,L²). A pair of fuzzy multifunctions ϕ = hϕL, ϕRiof 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)≤¹λR(↑²(ϕL(o1))(a2)).

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It is not difficult to check that the definition ofλ-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. Mappingid^X defined byid^X(x) =x for any x∈X is said to be an identity mapping on X.

Lemma 8. Any L-Chu correspondence is ahid^L,id^Li-Chu correspondence.

We are now in position to define the category of parameterized fuzzy formal contexts andλ-Chu correspondences between them:

Definition 14. We introduce a new category whose objects are parameterized fuzzy formal contextshB, A,L, riandλ-Chu correspondences between them.

The identity arrowof an object hB, A,L, riis the hid^L,id^Li-Chu correspon- denceι such that

– ιL(o) =↓↑(χo)for anyo∈B – ιR(a) =↑↓(χa)for any a∈A

– where χx(x) = 1and χx(y) = 0 for anyy6=x.

Composition of arrows³ hλ, ϕi: C¹ → C² and hµ, ψi: C² → C³ where Cⁱ = hBi, Ai,Lⁱ, riifori∈ {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) = _

o2∈B2

ψ(o2)(o3)⊗³µL(ϕL(o1)(o2)) ϕR+(ψR(a3))(a1) = _

a2∈A2

ϕR(a2)(a1)⊗¹λR(ψR(a3)(a2))

Obviously, one has to check that the proposed notions of composition and identity 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 hid^L,id^Li-Chu correspondence.

Lemma 10. Consider hλ, ϕi:C¹→ C² andhµ, ψi:C²→ C³, thenhµ, ψi ◦ hλ, ϕi is a(µ◦λ)-Chu correspondence. Moreover, composition ofλ-correspondences is associative.

3 Anyλ-Chu correspondenceϕcan be conveniently denoted byhλ, ϕi.

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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 ωⁱ for i ∈ {1,2} are defined by ωⁱ(x)(y) = ki, provided that ω(x)(y) = (k1, k2).

Transposition of such multifunction is defined byω^t(y)(x) =ω(x)(y).

Definition 16. Given two fuzzy formal contextsCⁱ=hBi, Ai,Lⁱ, rii,i∈ {1,2}, and λ∈CRL(L¹,L²). A λ-bond is a multifunctionβ: B1 →(L1×L2)^A² such that, for any (o1, a2)∈B1×A2:

β²(o1)is an intent ofC² (11)

(β^t)¹(a2) is an extent ofC¹ (12)

β¹(o1)(a2)≤¹λR(β²(o1)(a2))or equivalentlyλL(β¹(o1)(a2))≤²β²(o1)(a2) (13) The known relation between L-bonds and L-Chu correspondences is pre- served in theλ-case. Formally,

Lemma 11. Let β be a λ-bond between two fuzzy contextsCⁱ =hBi, Ai,Lⁱ, rii fori∈ {1,2}. Thenϕβ defined as

ϕβL(o1) = ↓²(β²(o1)) (14) ϕβR(a2) = ↑¹((β^t)¹(a2)) (15) isλ-Chu correspondence.

Proof. By calculation

↑²(ϕβL(o1))(a2)⁽¹⁴⁾= ↑²↓²(β²(o1))(a2)⁽¹¹⁾= β²(o1)(a2)

(13)

≥ λL(β¹(o1)(a2)) =λL((β^t)¹(a2)(o1))

(12)= λL(↓¹↑¹((β^t)¹(a2))(o1))

(15)= λL(↓¹(ϕβR(a2))(o1))

u t Lemma 12. Let L¹,L²be two complete residuated lattices satisfying the double negation law and let λ∈CRL(L¹,L²). ThenΦλ satisfies double negation law.

Proof. Consider an arbitraryhk1, k2i ∈FPλ. We have that, by definition,

¬¬hk1, k2i= (hk1, k2i →0)→0 = (hk1, k2i → hλR(02),02i)→ hλR(02),02i

=h(k1→¹λR(02))→¹λR(02),(k2→²02)→²02i

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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→²02)→²02

= k2→²λLλR(02)

→²λLλR(02)

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≥²λL(λRλL(λR(k2)→¹λR(02))→¹λR(02))

(?)= λL((λR(k2)→¹λR(02))→¹λR(02))

=λL((k1→¹λR(02))→¹λR(02))

(∗)

≥2λL(k1)

where equality (?) follows because, by Lemma 7,λR(k2)→¹λR(02) is a closed value inL1for the compositionλRλL, and inequality (∗) follows from the mono- tonicity ofλL.

As a result of the previous chain we obtain the following equality λL(k1) =λL((k1→¹λR(02))→¹λR(02))

and, again by Lemma 7, since k1 and λR(02) are closed for λRλL, as a result (k1→¹λR(02))→¹λR(02) is closed too, andk1= (k1→¹λR(02))→¹λR(02).

u t We are now ready to include the characterization result on the structure of λ-bonds, but we have to introduce the notion ofλ-direct product of contexts.

Definition 17. Let Cⁱ = hBi, Ai,Lⁱ, rii 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, ∆λi where ∆λ((o1, a2),(o2, a1)) =

¬(λ1(r1))→λ2(r2(o2, a2))is said to be the λ-direct productofC¹andC², where λ1(k) =hλRλL(k), λL(k)ifor allk∈L1 (16) λ2(k) =hλR(k), λLλR(k)ifor allk∈L2 (17) Lemma 13. Let C¹∆λC² be the λ-direct product of fuzzy formal contexts C¹ and C², and λ∈CRL(L¹,L²). For any extent of C¹∆λC² there exists a λ-bond between C¹ andC².

Proof. Lethβ, γibe a concept ofC¹∆λC². Thenβ ∈FP^B_λ¹^×^A² ⊆(L1×L2)^B¹^×^A². β(o1, a2) =↓^∆^λ (γ)(o1, a2)

= ^

o2∈B2

^

a1∈A1

(γ(o2, a1)→∆λ((o1, a2),(o2, a1)))

= ^

o2∈B2

^

a1∈A1

(γ(o2, a1)→(¬λ1(r1(o1, a1)))→λ2(r2(o2, a2)))

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Then

β¹(o1, a2) = ^

o2∈B2

^

a1∈A1

(γ¹(o2, a1)→¹(¬λRλL(r1(o1, a1)))→¹λR(r2(o2, a2)))

= ^

o2∈B2

^

a1∈A1

((γ¹(o2, a1)⊗¹¬λRλL(r1(o1, a1)))→¹λR(r2(o2, a2)))

(7)= ^

o2∈B2

^

a1∈A1

λR(λL(γ¹(o2, a1)⊗¹¬λRλL(r1(o1, a1)))→²r2(o2, a2))

=λR( ^

o2∈B2

^

a1∈A1

(λL(γ¹(o2, a1)⊗¹¬λRλL(r1(o1, a1)))→²r2(o2, a2)))

=λR( ^

o2∈B2

( _

a1∈A1

(λL(γ¹(o2, a1)⊗¹¬λRλL(r1(o1, a1))))→²r2(o2, a2)))

=λR( ^

o2∈B2

(σ(o1)(o2)→²r2(o2, a2)))

=λR(↑²(σ(o1))(a2)) Similarly

β²(o1, a2) = ^

o2∈B2

^

a1∈A1

((γ²(o2, a1)⊗²¬λLλR(r2(o2, a2)))→²λL(r1(o1, a1)))

(6)

≥² ^

o2∈B2

^

a1∈A1

λL(λR(γ²(o2, a1)⊗²¬λLλR(r2(o2, a2)))→²r1(o1, a1))

≥²λL( ^

o2∈B2

^

a1∈A1

(λR(γ²(o2, a1)⊗²¬λLλR(r2(o2, a2)))→¹r1(o1, a1)))

=λL( ^

a1∈A1

( _

o2∈B2

(λR(γ²(o2, a1)⊗²¬λLλR(r2(o2, a2))))→¹r1(o1, a1)))

=λL( ^

a1∈A1

(τ(a2)(a1)→¹r1(o1, a1))) =λL(↓¹(τ(a2))(o1)) Then let us define a multifunctionβb: B1−→(L1×L2)^A² as follows

βb²(o1) =↑²(σ(o1)) (βb^t)¹(a2) =↓¹(τ(a2))

whereσandτ are multifunctions above. We see thatβb²(o1) is the intent ofC², (βb^t)¹(a2) is the extent ofC¹and moreoverβ(o1, a2)∈FPλ, henceλL(β¹(o1, a2)) = β²(o1, a2) and

λL((βb^t)¹(a2)(o1)) =λL(↓¹(τ(a2)))

≤²β²(o1, a2)

=λL(β¹(o1, a2))

=λLλR(↑²(σ(o1))(a2))

≤²↑²(σ(o1))(a2) =βb²(o1)(a2)

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Thereforeβbis aλ-bond betweenC¹andC². ut

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.

S Math Phi Chem Bio

Anna A A B C

Boris C B A B

Cyril D E B C

U CS Tech Med

Math Ex G W

Phi G Ex G

Chem W Ex Ex

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 λL(−) Ex G W W W W

Ex G W λR(−) A B C

hλL, λRi is an isotone Galois connection. In fact, hλL, λRi is a residuation- preserving isotone Galois connection over Lukasiewicz logic, whose set of fixpoints is

FPλ={(A,Ex); (B,G); (C,W)}

Theλ-direct productS∆λUis 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 ofS∆λU 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.

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Med Tech CS Anna B,G B,G A,Ex Boris A,Ex B,G C,W Cyril B,G B,G C,W

Math Phi Chem Bio Math A,Ex A,Ex B,G C,W Phi B,G A,Ex A,Ex B,G Chem C,W B,G A,Ex B,G Bio C,W B,G A,Ex B,G Such concept should be translated into{1; 0.5; 0}structure.

Med Tech CS Anna 0.5 0.5 1 Boris 1 0.5 0 Cyril 0.5 0.5 0

Math Phi Chem Bio

Math 1 1 0.5 0

Phi 0.5 1 1 0.5

Chem 0 0.5 1 0.5

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 theλ-connection between such truth degrees structures as follows:

A B C D E F λL(−) Ex G G W W W

Ex G 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.

Med Tech CS Anna B,G A,Ex A,Ex Boris A,Ex A,Ex B,G Cyril B,G B,G D,W

Math Phi Chem Bio Math A,Ex A,Ex B,G B,G Phi A,Ex A,Ex A,Ex A,Ex Chem B,G B,G A,Ex B,G Bio B,G B,G B,G B,G

(12)

Translating the context into{1; 0.5; 0}structure, we obtain Med Tech CS

Anna 0.5 1 1 Boris 1 1 0.5 Cyril 0.5 0.5 0

Math Phi Chem Bio

Math 1 1 0.5 0.5

Phi 1 1 1 0.5

Chem 0.5 0.5 1 0.5 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 ofL-Chu correspondence by introducing the category CRL-ChuCors incorporating residuation to the underlying complete lattice L, specifically, on the basis of a residuation-preserving isotone Galois connectionλ. Then, theL-bonds are generalized within this same framework, and its structure is related to that of the extent of a suitably defined λ-direct product. A first relationship between extents ofλ-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 betweenC¹andC².

Potential applications are primary motivations for further future work, for instance, to consider possible classes of formalL-contexts induced from existing datamining notions, and study its associated categories.

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