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. References 1. J. T. Denniston, A. Melton, and S. E. Rodabaugh. Formal concept analysis and lattice-valued Chu systems. Fuzzy Sets and Systems, 216:52–90, 2013. 2. P. Hitzler and G.-Q. Zhang. A cartesian closed category of approximable concept structures. Lecture Notes in Computer Science, 3127:170–185, 2004. 3. S. Krajči. A categorical view at generalized concept lattices. Kybernetika, 43(2):255–264, 2007. 4. O. Krı́dlo, S. Krajči, and M. Ojeda-Aciego. The category of L-Chu correspondences and the structure of L-bonds. 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