Linking L-Chu correspondences and completely lattice L-ordered sets

Transcription

1 Linking L-Chu correspondences and completely lattice L-ordered sets Ondrej Krídlo 1 and Manuel Ojeda-Aciego 2 1 University of Pavol Jozef Šafárik, Košice, Slovakia 2 Dept. Matemática Aplicada, Univ. Málaga, Spain Abstract. Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets. 1 Introduction This paper deals with an extremely general form of Formal Concept Analysis FCA) based on categorical constructs and L-fuzzy sets. FCA has become an extremely useful theoretical and practical tool for formally describing structural and hierarchical properties of data with object-attribute character, and this applicability justifies the need of a deeper knowledge of its underlying mechanisms: and one important way to obtain this extra knowledge turns out to be via generalization and abstraction. Several approaches have been presented for generalizing the framework and the scope of formal concept analysis and, nowadays, one can see works which extend the theory by using ideas from fuzzy set theory, rough set theory, or possibility theory [1, 10, 18, 20 22, 24]. Concerning applications of fuzzy formal concept analysis, one can see papers ranging from ontology merging [9], to applications to the Semantic Web by using the notion of concept similarity or rough sets [11,12], and from noise control in document classification [19] to the development of recommender systems [7]. We are concerned in this work with the category L-ChuCors, built on top of several fuzzy extensions of the classical concept lattice, mainly introduced by Bělohlávek [3, 5, 6], who extended the underlying interpretation on classical logic to the more general framework of L-fuzzy logic [13]. The categorical treatment of morphisms as fundamental structural properties has been advocated by [17] as a means for the modelling of data translation, communication, and distributed computing, among other applications. Our approach broadly continues the research line which links the theory of Chu spaces with concept lattices [25] but, particularly, is based on the notion of Chu correspondences between formal contexts developed by Mori in [23]. Previous work Partially supported by grant VEGA 1/0832/12 and APVV Partially supported by Spanish Ministry of Science and FEDER funds through project TIN C05-01 and Junta de AndalucÃa project P09-FQM c 2012 by the paper authors. CLA 2012, pp Copying permitted only for private and academic purposes. Volume published and copyrighted by its editors. Local Proceedings in ISBN , Universidad de Málaga Dept. Matemática Aplicada), Spain.

2 234 Ondrej Krídlo and Manuel Ojeda-Aciego in this categorical approach has been developed by the authors in [14, 16]. The category L-ChuCors is formed by considering the class of L-contexts as objects and the L-fuzzy Chu correspondences as arrows between objects. Recently, the authors developed a further abstraction [15] aiming at formally describing structural properties of intercontextual relationships of L-contexts. The main result in this work is a constructive proof of the equivalence between the categories of L-formal contexts and L-Chu correspondences and that of completely lattice L-ordered sets and their corresponding morphisms. In order to obtain a reasonably self-contained document, Section 2 introduces the basic definitions concerning the L-fuzzy extension of formal concept analysis, as well as those concerning L-Chu correspondences; then, the categories associated to L-formal contexts and L-CLLOS are defined in Section 3 and, finally, the proof of equivalence is in Section 4. 2 Preliminaries 2.1 Basics of L-fuzzy FCA Definition 1. An algebra L,,,,,0,1 is said to be a complete residuated lattice if L,,,0,1 is a complete lattice with the least element 0 and the greatest element 1, L,,1 is a commutative monoid, and are adjoint, i.e. a b c if and only if a b c, for all a,b,c L, where is the ordering in the lattice generated from and. Definition 2. Let L be a complete residuated lattice, an L-fuzzy context is a triple B,A,r consisting of a set of objects B, a set of attributes A and an L- fuzzy binary relation r, i.e. a mapping r: B A L, which can be alternatively understood as an L-fuzzy subset of B A. Definition 3. Consider an L-fuzzy context B,A,r. Mappings : L B L A and : L A L B can be defined for every f L B and g L A as follows: f)a) ) fo) ro,a) g)o) ) ga) ro,a) Definition 4. An L-fuzzy concept is a pair f,g such that f) = g and g) = f. The first component f is said to be the extent of the concept, whereas the second component g is the intent of the concept. The set of all L-fuzzy concepts associated to a fuzzy context B,A,r will be denoted as L-FCLB, A, r). An ordering between L-fuzzy concepts is defined as follows: f 1,g 1 f 2,g 2 if and only if f 1 f 2 f 1 o) f 2 o) for all o B) if and only if g 1 g 2 g 1 o) g 2 o) for all a A). a A

3 Linking L-Chu correspondences and completely lattice L-ordered sets 235 Theorem 1 See [5]). The poset L-FCLB,A,r), ) is a complete lattice where f j,g j = f j, f j ) j J j J j J f j,g j = j ), j J j Jg g j j J Moreover a complete lattice V = V, is isomorphic to L-FCLB,A,r) iff there are mappings γ : B L V and µ : A L V, such that γb L) is -dense and µa L is -dense in V, and k l) ro,a) is equivalent to γo,k) µa,l) for all o B, a A and k,l L. Bělohlávek has extended the fundamental theorem of concept lattices by Dedekind-MacNeille completion in fuzzy settings by using the notions of L- equality and L-ordering. All the definitions and related constructions given until the end of the section are from [6]. Definition 5. A binary L-relation on X is called an L-equality if it satisfies 1. x x) = 1, reflexivity), 2. x y) = y x), symmetry), 3. x y) y z) x z), transitivity), 4. x y) = 1 implies x = y L-equality is a natural generalization of the classical bivalent) notion. Definition 6. An L-ordering or fuzzy ordering) on a set X endowed with an L-equality relation is a binary L-relation which is compatible w.r.t. i.e. fx) x y) fy), for all x,y X) and satisfies 1. x x = 1, reflexivity), 2. x y) y x) x y), antisymmetry), 3. x y) y z) x z), transitivity). If is an L-order on a set X with an L-equality, we call the pair X, an L-ordered set. Clearly, if L = 2, the notion of L-order coincides with the usual notion of partial) order. Definition 7. An L-set f L X is said to be an L-singleton in X, if it is compatible w.r.t. and the following holds: 1. there exists x 0 X with fx 0 ) = 1 2. fx) fy) x y), for all x,y X. Definition 8. For an L-ordered set X, and f L X we define the L-sets inff) and supf) in X by

4 236 Ondrej Krídlo and Manuel Ojeda-Aciego 1. inff)x) = Lf))x) ULf))x) 2. supf)x) = Uf))x) LUf))x) where Lf)x) ) y X fy) x y) Uf)x) ) y X fy) y x) inff) and supf) are called infimum or supremum, respectively. Definition 9. An L-ordered set X, is said to be completely lattice L-ordered set if for any f L X both supf) and inff) are -singletons. By proving of all the following lemmas some of the properties of residuated lattices are used. All details could be found in [4]. Some of needed properties are listed below. 1. k l m)) = k l) m) = l k) m) = l k m)) 2. k i I m i i I k m i) 3. i I m i) k i I m i k) Lemma 1. For any pair of L-concepts f i,g i L-FCLB,A,r) i {1,2}) of any L-context B, A, r the following equality holds. f1 o) f 2 o) ) g2 a) g 1 a) ) a A Proof. f1 o) f 2 o) ) f1 o) g 2 )o) ) f1 o) g2 a) ro,a) )) a A g2 a) f1 o) ro,a) )) a A g2 a) f 1 )a) ) a A g2 a) g 1 a) ) a A Definition 10. We define an L-equality and L-ordering on the set of formal concepts L-FCLC) of L-context C as follows: 1. f 1,g 1 f 2,g 2 f1 o) f 2 o) ) a A g2 a) g 1 a) ) 2. f 1,g 1 f 2,g 2 f1 o) f 2 o) ) a A g2 a) g 1 a) ) where k m is defined as k m) m k) for any k,m L.

5 Linking L-Chu correspondences and completely lattice L-ordered sets 237 Definition 11. Let C = B,A,r be an L-fuzzy formal context and γ be an L- set from L L-FCLC). We define L-sets of objects and attributes B γ and A γ, respectively, as follows: 1. B γ)o) = γ f,g ) fo)), for o B f,g L-FCLC) 2. A γ)a) = f,g L-FCLC) γ f,g ) ga)), for a A Theorem 2. Let C = B,A,r be an L-context. L-FCLC),, is a completely lattice L-ordered set in which infima and suprema can be described as follows: for an L-set γ L L FCLC) we have: { 1 infγ) = γ ), γ ) } { 1 supγ) = γ ), γ ) } A A Moreover a completely lattice L-ordered set V = V,, is isomorphic to L-FCL B,A,r ), 1, 1 iff there are mappings γ : B L V and µ : A L V, such that γb L) is {0,1}-supremum dense and µa L) is {0,1}- infimum dense in V, and k l) ro,a)) = γo,k) µa,l)) for all o B, a A and k,l L. In particular, V is isomorphic to L-FCLV,V, ), 1, 1. B B 2.2 L-Chu correspondences Definition 12. Consider two L-fuzzy contexts C i = B i,a i,r i,i = 1,2), then the pair ϕ = ϕ L,ϕ R ) is called a correspondence from C 1 to C 2 if ϕ L and ϕ R are L-multifunctions, respectively, from B 1 to B 2 and from A 2 to A 1 that is, ϕ L : B 1 L B2 and ϕ R : A 2 L A1 ). The L-correspondence ϕ is said to be a weak L-Chu correspondence if the equality ϕ R a 2 )a 1 ) r 1 o 1,a 1 )) = ϕ L o 1 )o 2 ) r 2 o 2,a 2 )) 1) holds for all o 1 B 1 and a 2 A 2. A weak Chu correspondence ϕ is an L-Chu correspondence if ϕ L o 1 ) is an L-set of objects closed in C 2 and ϕ R a 2 ) is an L-set of attributes closed in C 1 for all o 1 B 1 and a 2 A 2. We will denote the set of all L-Chu correspondences from C 1 to C 2 by L-ChuCorsC 1,C 2 ). Definition 13. Given a mapping : X L Y, we define +: L X L Y for all f L X by +f)y) = ) x X fx) x)y). 3 Introducing the relevant categories 3.1 The category L-ChuCors objects L-fuzzy formal contexts

6 238 Ondrej Krídlo and Manuel Ojeda-Aciego arrows L-Chu correspondences identity arrow ι : C C of L-context C = B,A,r ι L o) = χ o ), for all o B ι R a) = χ a ), for all a A composition ϕ 2 ϕ 1 : C 1 C 3 of arrows ϕ 1 : C 1 C 2, ϕ 2 : C 2 C 3 C i = B i,a i,r i, i {1,2}) ϕ 2 ϕ 1 ) L : B 1 L B3 defined as ϕ 2 ϕ 1 ) L o 1 ) = 3 3 ϕ2l+ ϕ 1L o 1 )) ) where ϕ 2L+ ϕ 1L o 1 ))o 3 ) = ϕ 1L o 1 )o 2 ) ϕ 2L o 2 )o 3 ) andϕ 2 ϕ 1 ) R : A 3 L A1 definedasϕ 2 ϕ 1 ) R a 3 ) = 1 1 ϕ1r+ ϕ 2R a 3 )) ) where ϕ 1R+ ϕ 2R a 3 ))a 1 ) = ϕ 2R a 3 )a 2 ) ϕ 1R a 2 )a 1 ) All details about definition of the category L-ChuCors could be found in [15]. 3.2 Category L-CLLOS Here we define another category Objects are completely lattice L-ordered sets in short, L-CLLOS) i.e. our objects will be represented as V = V,, Arrows are pairs of mappings between two L- CLLOSs i.e. s,z between V 1 and V 2, such that: 1. s : V 1 V 2, 2. z : V 2 V 1, 3. sv 1 ) 2 v 2 ) = v 1 1 zv 2 )), for all v 1,v 2 ) V 1 V 2. Identity arrow of V,, is a pair of identity morphisms id V,id V Composition of arrows is based on composition of mappings: consider two arrows s i,z i : V i V i+1, where i {1,2}. Composition is defined as follows: s 2,z 2 s 1,z 1 = s 2 s 1,z 1 z 2. Thus, given a pair of two arbitrary elements v 1,v 3 ) V 1 V 3 then: s2 s 1 )v 1 ) 3 v 3 ) = s2 s 1 v 1 )) 3 v 3 ) = s 1 v 1 ) 2 z 2 v 3 ) ) = v 1 1 z 1 z 2 v 3 )) ) = v 1 1 z 1 z 2 )v 3 ) ) Associativity of composition follows trivially because of the associativity of composition of mappings between sets.

7 Linking L-Chu correspondences and completely lattice L-ordered sets The categories L-ChuCors and L-CLLOS are equivalent In this section we start to build the equivalence by introducing a functor Γ from L-ChuCors to L-CLLOS in the following way: 1. ΓC) = L-FCLC),, for any L-context C will be its L-concept L- CLLOS. 2. Γϕ) = ϕ,ϕ. To any L-Chu correspondence ϕ L-ChuCorsC 1,C 2 ), Γϕ) will be a pair of mappings ϕ,ϕ defined as follows: ϕ f1,g 1 ) = 2 2 ϕl+ f 1 ) ), 2 ϕl+ f 1 ) ) ϕ f2,g 2 ) = 1 ϕr+ g 2 ) ), 1 1 ϕr+ g 2 ) ) where f i,g i L-FCLC i ) for i {1,2}. Lemma 2. Γϕ) L-CLLOSΓC 1 ),ΓC 2 )) for any ϕ L-ChuCorsC 1,C 2 ). Proof. Consider two arbitrary L-concepts f i,g i of L-FCLC i ), i, i for i {1,2}, such that C i = B i,a i,r i. ϕ f1,g 1 ) 2 f 2,g 2 = 2 2 ϕl+ f 1 ) ), 2 ϕl+ f 1 ) ) 2 f 2,g 2 g2 a 2 ) 2 ϕl+ f 1 ) ) a 2 ) ) o 1 B 1 g2 a 2 ) o 1 B 1 o 1 B 1 o 1 B 1 o 1 B 1 g2 a 2 ) f 1 o 1 ) ϕl o 1 )o 2 ) f 1 o 1 ) ) r 2 o 2,a 2 ) )) f1 o 1 ) g 2 a 2 ) f1 o 1 ) f1 o 1 ) 1 ϕr+ g 2 ) ) o 1 ) ) = f 1,g ϕr+ g 2 ) ), 1 1 ϕr+ g 2 ) ) ϕl o 1 )o 2 ) r 2 o 2,a 2 ) ))) ϕr a 2 )a 1 ) r 1 o 1,a 1 ) ))) ϕr a 2 )a 1 ) g 2 a 2 ) ) r 1 o 1,a 1 ) )) = f 1,g 1 1 ϕ f2,g 2 ) Lemma 3. For the identity arrow ι L-ChuCorsC,C) of any L-context C = B,A,r, Γι) is the identity arrow from L-CLLOSΓC),ΓC)). Proof. Consider any L-concept f, g from L-FCLC). ι L+ f) ) a) ιl+ f)o) ro,a) ) ιl b)o) fb) ) ro,a) ) b B

8 240 Ondrej Krídlo and Manuel Ojeda-Aciego ιl b)o) fb) ) ro,a) ) b B fb) ιl b)o) ro,a) ) b B ) ) fb) χb a) b B ) fb) rb,a) = f)a) b B Therefore, we have ι f,g ) = f,g. The proof for ι is similar. Lemma 4. Consider arbitrary ϕ i L-ChuCorsC i,c i+1 ) for i {1,2} and any element o 1 B 1 and g 3 L A3. Then Proof. 1 ϕ1r+ ϕ2r+ g 3 ) )) o 1 ) = 1 ϕ1r+ 2 2 ϕ2r+ g 3 ) ))) o 1 ). 1 ϕ1r+ ϕ2r+ g 3 ) )) o 1 ) ϕ1r+ ϕ2r+ g 3 ) ) a 1 ) r 1 o 1,a 1 ) ) ϕ1r a 2 )a 1 ) ϕ 2R+ g 3 )a 2 ) ) r 1 o 1,a 1 ) ) ϕ2r+ g 3 )a 2 ) ϕ2r+ g 3 )a 2 ) ϕ1l o 1 )o 2 ) ϕ1r a 2 )a 1 ) r 1 o 1,a 1 ) )) ϕ1l o 1 )o 2 ) r 2 o 2,a 2 ) )) ϕ2r+ g 3 )a 2 ) r 2 o 2,a 2 ) )) ϕ1l o 1 )o 2 ) ϕ2r+ g 3 ) ) o 2 ) ) by applying the same chain of modifications in opposite way we will obtain = 1 ϕ1r+ 2 2 ϕ2r+ g 3 ) ))) o 1 ) Lemma 5. Mapping Γ is closed under arrow composition. Proof. Considerϕ i L-ChuCorsC i,c i+1 )fori {1,2}.Let f i,g i L-FCLC i ) be an arbitrary L-context for all i {1,3}. Recall that 1. Γϕ 2 ϕ 1 ) = ϕ 2 ϕ 1 ),ϕ 2 ϕ 1 ) 2. Γϕ 2 ) Γϕ 1 ) = ϕ 2 ϕ 1,ϕ 1 ϕ 2

9 Linking L-Chu correspondences and completely lattice L-ordered sets 241 The proof will be based on equality of corresponding elements of the previous pairs: only one part will be proved, the other one is similar. ϕ 1 ϕ 2 ) f 3,g 3 ) = ϕ 1 ϕ2 f3,g 3 )) = 1 ϕ 1R+ 2 2 ϕ 2R+ g 3 )))), 1 1 ϕ 1R+ 2 2 ϕ 2R+ g 3 )))) by lemma 4 we have = 1 ϕ 1R+ ϕ 2R+ g 3 ))), 1 1 ϕ 1R+ ϕ 2R+ g 3 ))) = 1 ϕ 1R+ ϕ 2R+ g 3 )))o 1 ) = ) ϕ 1R a 2 )a 1 ) ϕ 2R+ g 3 )a 2 )) r 1 o 1,a 1 ) ) ϕ 1R a 2 )a 1 ) ϕ 2R a 3 )a 2 ) g 3 a 3 ))) r 1 o 1,a 1 ) a 3 A 3 ϕ1r+ ϕ 2R a 3 ))a 1 ) g 3 a 3 ) ) r 1 o 1,a 1 ) ) a 3 A 3 a 3 A 3 a 3 A 3 a 3 A 3 a 3 A 3 a 3 A 3 g3 a 3 ) ϕ1r+ ϕ 2R a 3 ))a 1 ) r 1 o 1,a 1 ) )) g3 a 3 ) ϕ1r+ ϕ 2R a 3 )) ) o 1 ) ) g3 a 3 ) 1 ϕ2 ϕ 1 ) R a 3 ) ) o 1 ) ) g3 a 3 ) ϕ2 ϕ 1 ) R a 3 )a 1 ) r 1 o 1,a 1 ) )) g3 a 3 ) ϕ 2 ϕ 1 ) R a 3 )a 1 ) ) r 1 o 1,a 1 ) ) ) g 3 a 3 ) ϕ 2 ϕ 1 ) R a 3 )a 1 )) r 1 o 1,a 1 ) a 3 A 3 = 1 ϕ2 ϕ 1 ) R+ g 3 ) ) o 1 ) = 1 ϕ 2 ϕ 1 ) R+ g 3 )), 1 1 ϕ 2 ϕ 1 ) R+ g 3 )) =ϕ 2 ϕ 1 ) f 3,g 3 ) Proposition 1. Γ is a functor from L-ChuCorrs to L-CLLOS. Proof. Straightforward from the previous lemmas. We continue by showing that the previous functor satisfies the conditions to define a categorical equivalence, characterized by the following result:

10 242 Ondrej Krídlo and Manuel Ojeda-Aciego Theorem 3 See [2]). The following conditions on a functor F : C D are equivalent: F is an equivalence of categories. F is full and faithful and essentially surjective on objects: for every D D there is some C C such that FC) = D. Let us recall the definition of the notions required by the previous theorem: Definition A functor F : C D is faithful if for all objects A,B of a category C, the map F A,B : Hom C A,B) Hom D FA),FB)) is injective. 2. Similarly, F is full if F A,B is always surjective. In our cases, for proving fullness and faithfulness of the functor Γ we need to prove surjectivity and injectivity of the mapping Γ C1,C 2 : L-ChuCorsC 1,C 2 ) L-CLLOSΓC 1 ),ΓC 2 )) for any two L-contexts C 1 and C 2. This will be done in the forthcoming lemmas. Lemma 6. Γ is full. Proof. The point of the proof is to show that given any arrow s,z from the set L-CLLOSΓC 1 ),ΓC 2 )) there exists an L-Chu correspondence ϕ s,z from the set L-ChuCorsC 1,C 2 ), for any two L-contexts C i = B i,a i,r i for i = {1,2}. Let us define the following mappings: ϕ s,z L o 1 ) = Ext s 1 1 χ o1 ), 1 χ o1 ) )) ϕ s,z R a 2) = Int z 2 χ a2 ), 2 2 χ a2 ) )) s,z 2 ϕ L o 1 ) ) a 2 ) ϕ s,z L o 1 )o 2 ) r 2 o 2,a 2 ) ) Exts 1 1 χ o1 ), 1 χ o1 ) ))o 2 ) 2 χ a2 )o 2 ) ) = s 1 1 χ o1 ), 1 χ o1 ) ) 2 2 χ a2 ), 2 2 χ a2 ) = 1 1 χ o1 ), 1 χ o1 ) 1 z 2 χ a2 ), 2 2 χ a2 ) ) Intz 2 χ a2 ), 2 2 χ a2 ) ))a 1 ) 1 χ o1 )a 1 ) ) ϕ s,z R a 2)a 1 ) r 1 o 1,a 1 ) ) = 1 ϕ s,z R a 2) ) o 1 ) So ϕ s,z L-ChuCorsC 1,C 2 ) and Γ C1,C 2 is surjective, hence Γ is full.

11 Linking L-Chu correspondences and completely lattice L-ordered sets 243 Lemma 7. Γ is faithfull Proof. Now the point is to prove the injectivity of Γ C1,C 2. Consider two L-Chu correspondences ϕ 1,ϕ 2 from L-ChuCorsC 1,C 2 ) such that ϕ 1 ϕ 2, and let us fix the pair o 1,a 2 ) B 1 A 2, such that 2 ϕ1l o 1 ) ) a 2 ) = 1 ϕ1r a 2 ) ) o 1 ) 2 ϕ2l o 1 ) ) a 2 ) = 1 ϕ2r a 2 ) ) o 1 ) Let us assume that either 1 ϕ1r a 2 ) ) o 1 ) > 2 ϕ2l o 1 ) ) a 2 ) or that both values from L are incomparable, that is equivalent to the following: 1 ϕ1r a 2 ) ) o 1 ) 2 ϕ2l o 1 ) ) a 2 ) < 1 Now consider the L-concept 1 ϕ 1R a 2 )),ϕ 1R a 2 ) and let us compare its images under the mappings ϕ 1 and ϕ 2. 2 ϕ2l+ 1 ϕ 1R a 2 )) )) a 2 ) ϕ2l+ 1 ϕ 1R a 2 )))o 2 ) r 2 o 2,a 2 ) ) b 1 B 1 b 1 B 1 b 1 B 1 ϕ2l b 1 )o 2 ) 1 ϕ 1R a 2 ))b 1 ) ) r 2 o 2,a 2 ) ) 1 ϕ 1R a 2 ))b 1 ) 1 ϕ 1R a 2 ))b 1 ) 2 ϕ 2L b 1 ))a 2 ) ) 1 ϕ 1R a 2 ))o 1 ) 2 ϕ 2L o 1 ))a 2 ) < 1 because of the restriction given above Similarly, we can obtain: ϕ2l b 1 )o 2 ) r 2 o 2,a 2 ) )) 2 ϕ 1L+ 1 ϕ 1R a 2 ))))a 2 ) = 1 ϕ 1R a 2 ))b 1 ) 2 ϕ 1L b 1 ))a 2 ) ) b 1 B 1 b 1 B 1 1 ϕ 1R a 2 ))b 1 ) 1 ϕ 1R a 2 ))b 1 ) ) = 1 It means that ϕ 1 1 ϕ 1R a 2 )),ϕ 1R a 2 ) ) ϕ 2 1 ϕ 1R a 2 )),ϕ 1R a 2 ) ) Hence ϕ 1 1 ϕ 1R a 2 )),ϕ 1R a 2 ) ) ϕ 2 1 ϕ 1R a 2 )),ϕ 1R a 2 ) ) and ϕ 1 ϕ 2. So Γ C1,C 2 is injective and Γ is faithfull. Proposition 2. The functor Γ is an equivalence functor between L-ChuCors and L-CLLOS. Proof. Fullness and faithfulness of Γ is given by previous lemmas. Essential surjectivity on objects is ensured by the fact that given any object V,, of L-CLLOSthereexistsanL-context V,V,,suchthatΓ V,V, ) = V,,. Hence, we can state that Γ is the functor of equivalence between L-ChuCors and L-CLLOS.

12 244 Ondrej Krídlo and Manuel Ojeda-Aciego References 1. C. Alcalde, A. Burusco, R. Fuentes-González, and I. Zubia. The use of linguistic variables and fuzzy propositions in the L-Fuzzy concept theory. Computers & Mathematics with Applications 628): , S. Awodey. Category Theory, Oxford Logic Guides 49, R. Bělohlávek. Fuzzy concepts and conceptual structures: induced similarities. In Joint Conference on Information Sciences, pages , R. Bělohlávek. Fuzzy Relational Systems: Foundation and Principles, Kluwer, R. Bělohlávek. Lattices of fixed points of fuzzy Galois connections. Mathematical Logic Quartely, 471): , R. Bělohlávek. Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic, 128: , P. du Boucher-Ryana and D. Bridge. Collaborative recommending using formal concept analysis. Knowledge-Based Systems, 195): , P. Burillo, R. Fuentes-González, L. González. On Completeness and Direction in Fuzzy Relational Systems. Mathware and Soft Computing, 5: , R.-C. Chen, C.-T. Bau, and C-J. Yeh Merging domain ontologies based on the WordNet system and Fuzzy Formal Concept Analysis techniques, Applied Soft Computing 112): , D. Dubois and H. Prade. Possibility theory and formal concept analysis: Characterizing independent sub-contexts, Fuzzy Sets and Systems 196: 4 16, A. Formica. Concept similarity in formal concept analysis: An information content approach. Knowledge-Based Systems, 211):80 87, A. Formica. Semantic Web search based on rough sets and Fuzzy Formal Concept Analysis. Knowledge-Based Systems, 26:40 47, J. Goguen. L-fuzzy sets. J. Math. Anal. Appl., 18: , S. Krajči. A categorical view at generalized concept lattices. Kybernetika, 432): , O. Krídlo, S. Krajči, and M. Ojeda-Aciego. The category of L-Chu correspondences and the structure of L-bonds. Fundamenta Informaticae, 1154): , O. Krídlo and M. Ojeda Aciego. On L-fuzzy Chu correspondences. Intl J of Computer Mathematics, 889): , M. Krötzsch, P. Hitzler, and G.-Q. Zhang. Morphisms in context. Lecture Notes in Computer Science, 3596: , Y. Lei and M. Luo. Rough concept lattices and domains. Annals of Pure and Applied Logic, 159: , S.-T. Li and F.C. Tsai. Noise control in document classification based on fuzzy formal concept analysis. Proc. of FUZZ-IEEE 11: , J. Medina. Multi-adjoint property-oriented and object-oriented concept lattices. Information Sciences, 190:95 106, J. Medina and M. Ojeda-Aciego. Multi-adjoint t-concept lattices. Information Sciences, 1805): , J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño. Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems, 1602): , H. Mori. Chu correspondences. Hokkaido Mathematical Journal, 37: , Q. Wu and Z. Liu. Real formal concept analysis based on grey-rough set theory. Knowledge-Based Systems, 221):38 45, G.-Q. Zhang. Chu spaces, concept lattices, and domains. Electronic Notes in Theoretical Computer Science, 83, 2004.