Linking L-Chu Correspondences and Completely Lattice L-ordered Sets

completely lattice L -ordered sets

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. Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category ofL-Chu correspondences betweenL-formal contexts and prove that it is equivalent to the category of completely latticeL-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 mecha- nisms: 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 Bˇelohl´avek [3,5,6], who extended the underlying interpretation on classical logic to the more general framework ofL-fuzzy logic [13].

The categorical treatment of morphisms as fundamental structural proper- ties has been advocated by [17] as a means for the modelling of data translation, communication, and distributed computing, among other applications. Our ap- proach 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 corre- spondences between formal contexts developed by Mori in [23]. Previous work

⋆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 ˜Aa project P09-FQM-5233.

c 2012 by the paper authors. CLA 2012, pp. 233–244. Copying permitted only for private and academic purposes. Volume published and copyrighted by its editors.

Local Proceedings in ISBN 978–84–695–5252–0,

Universidad de M´alaga (Dept. Matem´atica Aplicada), Spain.

in this categorical approach has been developed by the authors in [14, 16]. The categoryL-ChuCors is formed by considering the class ofL-contexts as objects and theL-fuzzy Chu correspondences as arrows between objects. Recently, the authors developed a further abstraction [15] aiming at formally describing struc- tural properties of intercontextual relationships ofL-contexts.

The main result in this work is a constructive proof of the equivalence be- tween the categories ofL-formal contexts andL-Chu correspondences and that of completely latticeL-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 andL-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 hL,∧,∨,⊗,→,0,1iis said to be a complete resid- uated latticeif

– hL,∧,∨,0,1i is a complete lattice with the least element0 and the greatest element 1,

– hL,⊗,1iis a commutative monoid,

– ⊗and→are adjoint, i.e.a⊗b≤cif and only ifa≤b→c, for alla, b, c∈L, where ≤is the ordering in the lattice generated from∧and∨.

Definition 2. Let Lbe a complete residuated lattice, an L-fuzzy contextis a triple hB, A, ri consisting of a set of objects B, a set of attributes Aand an L- fuzzy binary relationr, i.e. a mappingr:B×A→L, which can be alternatively understood as an L-fuzzy subset ofB×A.

Definition 3. Consider an L-fuzzy context hB, A, ri. Mappings ↑:L^B → L^A and↓:L^A→L^B can be defined for every f ∈L^B andg∈L^A as follows:

↑(f)(a) = ^

o∈B

f(o)→r(o, a)

↓(g)(o) = ^

a∈A

g(a)→r(o, a)

Definition 4. An L-fuzzy concept is a pair hf, gi such that ↑ (f) = g and

↓(g) =f. The first componentf is said to be theextentof the concept, whereas the second component g is theintentof the concept.

The set of all L-fuzzy concepts associated to a fuzzy context hB, A, riwill be denoted as L-FCL(B, A, r).

An ordering betweenL-fuzzy concepts is defined as follows:hf1, g1i ≤ hf2, g2i if and only if f1 ⊆ f2 (f1(o) ≤ f2(o) for all o ∈ B) if and only if g1 ⊇ g2

(g1(o)≥g2(o)for alla∈A).

Theorem 1 (See [5]). The poset (L-FCL(B, A, r),≤) is a complete lattice where

^

j∈J

hfj, gji=D ^

j∈J

fj,↑ ^

j∈J

fj)E _

j∈J

hfj, gji=D

↓ ^

j∈J

gj),^

j∈J

gj

E

Moreover a complete lattice V =hV,≤i is isomorphic to L-FCL(B, A, r)iff there are mappings γ : B×L → V and µ : A×L → V, such that γ(B×L) is W

-dense and µA×L isV

-dense in V, and (k⊗l)≤r(o, a) is equivalent to γ(o, k)≤µ(a, l) for allo∈B,a∈Aandk, l∈L.

Bˇelohl´avek has extended the fundamental theorem of concept lattices by Dedekind-MacNeille completion in fuzzy settings by using the notions of L- equality andL-ordering. All the definitions and related constructions given until the end of the section are from [6].

Definition 5. A binaryL-relation≈onX is called anL-equalityif it satisfies 1. (x≈x) = 1, (reflexivity),

2. (x≈y) = (y≈x), (symmetry),

3. (x≈y)⊗(y≈z)≤(x≈z), (transitivity), 4. (x≈y) = 1impliesx=y

L-equality is a natural generalization of the classical (bivalent) notion.

Definition 6. AnL-ordering(or fuzzy ordering) on a setX endowed with an L-equality relation ≈is a binaryL-relationwhich is compatible w.r.t.≈(i.e.

f(x)⊗(x≈y)≤f(y), for allx, y∈X) and satisfies 1. xx= 1, (reflexivity),

2. (xy)∧(yx)≤(x≈y), (antisymmetry), 3. (xy)⊗(yz)≤(xz), (transitivity).

If is anL-order on a set X with anL-equality≈, we call the pairhhX,≈i i anL-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 anL-singleton inhX,≈i if it is compatible w.r.t.≈and the following holds:

1. there existsx0∈X with f(x0) = 1 2. f(x)⊗f(y)≤(x≈y), for allx, y∈X.

Definition 8. For anL-ordered sethhX,≈i iandf ∈L^Xwe define theL-sets inf(f)andsup(f) inX by

1. inf(f)(x) = (L(f))(x)∧(UL(f))(x) 2. sup(f)(x) = (U(f))(x)∧(LU(f))(x) where

– L(f)(x) =V

y∈X f(y)→(xy) – U(f)(x) =V

y∈X f(y)→(yx)

inf(f)andsup(f) are called infimumor supremum, respectively.

Definition 9. An L-ordered set hhX,≈i i is said to be completely lattice L-ordered set if for anyf ∈L^X both sup(f)andinf(f)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→V

i∈Imi=V

i∈I(k→mi) 3. (W

i∈Imi)→k=V

i∈I(mi→k)

Lemma 1. For any pair ofL-concepts hfi, gii ∈L-FCL(B, A, r)(i∈ {1,2}) of any L-contexthB, A, rithe following equality holds.

^

o∈B

f1(o)→f2(o)

= ^

a∈A

g2(a)→g1(a)

Proof.

^

o∈B

f1(o)→f2(o)

= ^

o∈B

f1(o)→ ↓(g2)(o)

= ^

o∈B

f1(o)→ ^

a∈A

g2(a)→r(o, a)

= ^

a∈A

g2(a)→ ^

o∈B

f1(o)→r(o, a)

= ^

a∈A

g2(a)→ ↑(f1)(a)

= ^

a∈A

g2(a)→g1(a)

⊓

⊔

Definition 10. We define anL-equality≈andL-ordering on the set of for- mal conceptsL-FCL(C)of L-contextC as follows:

1. hf1, g1i hf2, g2i=V

o∈B f1(o)→f2(o)

=V

a∈A g2(a)→g1(a) 2. hf1, g1i ≈ hf2, g2i=V

o∈B f1(o)↔f2(o)

=V

a∈A g2(a)↔g1(a) wherek↔mis defined as (k→m)∧(m→k)for any k, m∈L.

Definition 11. Let C=hB, A, ribe an L-fuzzy formal context andγ be anL- set from L^L-^FCL(C). We define L-sets of objects and attributes S

Bγ andS

Aγ, respectively, as follows:

1. (S

Bγ)(o) = _

hf,gi∈L-^FCL(C)

(γ(hf, gi)⊗f(o)), for o∈B 2. (S

Aγ)(a) = _

hf,gi∈L-FCL(C)

(γ(hf, gi)⊗g(a)), fora∈A

Theorem 2. Let C=hB, A, ribe anL-context. hhL-FCL(C),≈i,i is a com- pletely lattice L-ordered set in which infima and suprema can be described as follows: for an L-setγ∈L^L−FCL(C) we have:

1inf(γ) =n

↓ [

A

γ

,↑↓ [

A

γo

1sup(γ) =n

↓↑ [

B

γ ,↑ [

B

γo

Moreover a completely lattice L-ordered set V = hhV,≈i,i is isomorphic to hhL-FCL(hB, A, ri),≈1i,1i iff there are mappings γ : B×L →V andµ : A×L→V, such thatγ(B×L)is{0,1}-supremum dense andµ(A×L)is{0,1}- infimum dense in V, and((k⊗l)→r(o, a)) = (γ(o, k)µ(a, l))for all o∈B, a∈Aandk, l∈L. In particular,V is isomorphic tohhL-FCL(V, V,),≈1i,1i.

2.2 L-Chu correspondences

Definition 12. Consider twoL-fuzzy contexts Ci=hBi, Ai, rii,(i= 1,2), then the pairϕ= (ϕL, ϕR)is called acorrespondence fromC1 toC2if ϕL andϕR

are L-multifunctions, respectively, from B1 to B2 and from A2 to A1 (that is, ϕL:B1→L^B2 andϕR:A2→L^A1).

The L-correspondence ϕ is said to be a weak L-Chu correspondence if the equality

^

a1∈A1

(ϕR(a2)(a1)→r1(o1, a1)) = ^

o2∈B2

(ϕL(o1)(o2)→r2(o2, a2)) (1)

holds for all o1∈B1 anda2∈A2.

A weak Chu correspondenceϕis anL-Chu correspondenceifϕL(o1)is an L-set of objects closed inC2andϕR(a2)is anL-set of attributes closed inC1for all o1 ∈B1 and a2 ∈A2. We will denote the set of all L-Chu correspondences fromC1 toC2 byL-ChuCors(C1, C2).

Definition 13. Given a mapping ̟:X →L^Y, we define ̟+: L^X → L^Y for allf ∈L^X by̟+(f)(y) =W

x∈X f(x)⊗̟(x)(y) .

3 Introducing the relevant categories

3.1 The category L-ChuCors – objects L-fuzzy formal contexts

– arrowsL-Chu correspondences

– identity arrowι:C→C ofL-contextC=hB, A, ri

• ιL(o) =↓↑(χo), for allo∈B

• ιR(a) =↑↓(χa), for alla∈A

– composition ϕ2◦ϕ1 : C1 →C3 of arrows ϕ1 :C1 →C2, ϕ2 :C2→ C3

(Ci=hBi, Ai, rii,i∈ {1,2})

• (ϕ2◦ϕ1)L:B1→L^B3 defined as (ϕ2◦ϕ1)L(o1) =↓3↑3 ϕ2L+(ϕ1L(o1)) where

ϕ2L+(ϕ1L(o1))(o3) = _

o2∈B2

ϕ1L(o1)(o2)⊗ϕ2L(o2)(o3)

• and (ϕ2◦ϕ1)R:A3→L^A1defined as (ϕ2◦ϕ1)R(a3) =↑1↓1 ϕ1R+(ϕ2R(a3)) where

ϕ1R+(ϕ2R(a3))(a1) = _

a2∈A2

ϕ2R(a3)(a2)⊗ϕ1R(a2)(a1)

All details about definition of the categoryL-ChuCors could be found in [15].

3.2 Category L-CLLOS Here we define another category

Objects are completely latticeL-ordered sets (in short,L-CLLOS) i.e. our ob- jects will be represented asV =hhV,≈i,i

Arrows are pairs of mappings between two L- CLLOSs i.e. hs, zi betweenV1

and V2, such that:

1. s:V1→V2, 2. z:V2→V1,

3. (s(v1)2v2) = (v11z(v2)), for all (v1, v2)∈V1×V2.

Identity arrow of hhV,≈i,i is a pair of identity morphismshidV, idVi Composition of arrows is based on composition of mappings: consider

two arrowshsi, zii:Vi→ Vi+1, where i∈ {1,2}. Composition is defined as follows:

hs2, z2i ◦ hs1, z1i=hs2◦s1, z1◦z2i.

Thus, given a pair of two arbitrary elements (v1, v3)∈V1×V3 then:

(s2◦s1)(v1)3v3

= s2(s1(v1))3v3

= s1(v1)2z2(v3)

= v11z1(z2(v3))

= v11(z1◦z2)(v3)

Associativity of composition follows trivially because of the associativ- ity of composition of mappings between sets.

4 The categories L-ChuCors and L-CLLOS are equivalent

In this section we start to build the equivalence by introducing a functorΓ from L-ChuCors toL-CLLOS in the following way:

1. Γ(C) = hhL-FCL(C),≈i,i for any L-context C will be its L-concept L- CLLOS.

2. Γ(ϕ) = hϕ_∨, ϕ_∧i. To any L-Chu correspondence ϕ ∈ L-ChuCors(C1, C2), Γ(ϕ) will be a pair of mappingshϕ_∨, ϕ_∧idefined as follows:

– ϕ∨ hf1, g1i

=

↓2↑2 ϕL+(f1)

,↑2 ϕL+(f1) – ϕ∧ hf2, g2i

=

↓1 ϕR+(g2)

,↑1↓1 ϕR+(g2) where hfi, gii ∈L-FCL(Ci) fori∈ {1,2}.

Lemma 2. Γ(ϕ)∈L-CLLOS(Γ(C1), Γ(C2))for anyϕ∈L-ChuCors(C1, C2).

Proof. Consider two arbitrary L-concepts hfi, gii of hhL-FCL(Ci),≈ii,ii for i∈ {1,2}, such thatCi=hBi, Ai, rii.

ϕ∨ hf1, g1i

2hf2, g2i

=

↓2↑2 ϕL+(f1)

,↑2 ϕL+(f1)

2hf2, g2i

= ^

a2∈A2

g2(a2)→ ↑2 ϕL+(f1) (a2)

= ^

a2∈A2

g2(a2)→ ^

o2∈B2

_

o1∈B1

ϕL(o1)(o2)⊗f1(o1)

→r2(o2, a2)

= ^

a2∈A2

^

o1∈B1

g2(a2)→ f1(o1)→ ^

o2∈B2

ϕL(o1)(o2)→r2(o2, a2)

= ^

a2∈A2

^

o1∈B1

f1(o1)→ g2(a2)→ ^

a1∈A1

ϕR(a2)(a1)→r1(o1, a1)

= ^

o1∈B1

f1(o1)→ ^

a1∈A1

_

a2∈A2

ϕR(a2)(a1)⊗g2(a2)

→r1(o1, a1)

= ^

o1∈B1

f1(o1)→ ↓1 ϕR+(g2) (o1)

=hf1, g1i 1

↓1 ϕR+(g2)

,↑1↓1 ϕR+(g2)

=hf1, g1i 1ϕ_∧ hf2, g2i

⊓

⊔ Lemma 3. For the identity arrow ι∈L-ChuCors(C, C) of any L-context C= hB, A, ri,Γ(ι)is the identity arrow fromL-CLLOS(Γ(C), Γ(C)).

Proof. Consider anyL-concepthf, gifromL-FCL(C).

↑ ιL+(f)

(a) = ^

o∈B

ιL+(f)(o)→r(o, a)

= ^

o∈B

_

b∈B

ιL(b)(o)⊗f(b)

→r(o, a)

= ^

o∈B

^

b∈B

ιL(b)(o)⊗f(b)

→r(o, a)

= ^

o∈B

f(b)→ ^

b∈B

ιL(b)(o)→r(o, a)

= ^

b∈B

f(b)→ ↑↓↑ χb

(a)

= ^

b∈B

f(b)→r(b, a)

=↑(f)(a)

Therefore, we haveι∨(hf, gi) =hf, gi. The proof forι∧ is similar. ⊓⊔ Lemma 4. Consider arbitraryϕi∈L-ChuCors(Ci, Ci+1)fori∈ {1,2}and any elemento1∈B1 andg3∈L^A3. Then

↓1 ϕ1R+ ϕ2R+(g3)

(o1) =↓1 ϕ1R+ ↑2↓2 ϕ2R+(g3) (o1).

Proof.

↓1 ϕ1R+ ϕ2R+(g3) (o1)

= ^

a1∈A1

ϕ1R+ ϕ2R+(g3)

(a1)→r1(o1, a1)

= ^

a1∈A1

_

a2∈A2

ϕ1R(a2)(a1)⊗ϕ2R+(g3)(a2)

→r1(o1, a1)

= ^

a2∈A2

ϕ2R+(g3)(a2)→ ^

a1∈A1

ϕ1R(a2)(a1)→r1(o1, a1)

= ^

a2∈A2

ϕ2R+(g3)(a2)→ ^

o2∈B2

ϕ1L(o1)(o2)→r2(o2, a2)

= ^

o2∈B2

ϕ1L(o1)(o2)→ ^

a2∈A2

ϕ2R+(g3)(a2)→r2(o2, a2)

= ^

o2∈B2

ϕ1L(o1)(o2)→ ↓2↑2↓2 ϕ2R+(g3) (o2)

by applying the same chain of modifications in opposite way we will obtain

=↓1 ϕ1R+ ↑2↓2 ϕ2R+(g3) (o1)

⊓

⊔

Lemma 5. MappingΓ is closed under arrow composition.

Proof. Considerϕi ∈L-ChuCors(Ci, Ci+1) fori∈ {1,2}. Lethfi, gii ∈L-FCL(Ci) be an arbitraryL-context for alli∈ {1,3}. Recall that

1. Γ(ϕ2◦ϕ1) =

(ϕ2◦ϕ1)_∨,(ϕ2◦ϕ1)_∧ 2. Γ(ϕ2)◦Γ(ϕ1) =

ϕ_2∨◦ϕ_1∨, ϕ_1∧◦ϕ_2∧

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.

(ϕ1∧◦ϕ2∧) hf3, g3i

=ϕ1∧ ϕ2∧ hf3, g3i

=

↓1(ϕ1R+(↑2↓2(ϕ2R+(g3)))),↑1↓1(ϕ1R+(↑2↓2(ϕ2R+(g3)))) by lemma 4 we have

=

↓1(ϕ1R+(ϕ2R+(g3))),↑1↓1(ϕ1R+(ϕ2R+(g3)))

=⋆

↓1(ϕ1R+(ϕ2R+(g3)))(o1) =

= ^

a1∈A1

_

a2∈A2

(ϕ1R(a2)(a1)⊗ϕ2R+(g3)(a2))→r1(o1, a1)

= ^

a1∈A1

_

a2∈A2

(ϕ1R(a2)(a1)⊗ _

a3∈A3

(ϕ2R(a3)(a2)⊗g3(a3)))→r1(o1, a1)

= ^

a1∈A1

_

a3∈A3

ϕ1R+(ϕ2R(a3))(a1)⊗g3(a3)

→r1(o1, a1)

= ^

a3∈A3

g3(a3)→ ^

a1∈A1

ϕ1R+(ϕ2R(a3))(a1)→r1(o1, a1)

= ^

a3∈A3

g3(a3)→↓1↑1↓1 ϕ1R+(ϕ2R(a3)) (o1)

= ^

a3∈A3

g3(a3)→↓1 (ϕ2◦ϕ1)R(a3) (o1)

= ^

a3∈A3

g3(a3)→ ^

a1∈A1

(ϕ2◦ϕ1)R(a3)(a1)→r1(o1, a1)

= ^

a1∈A1

^

a3∈A3

g3(a3)⊗(ϕ2◦ϕ1)R(a3)(a1)

→r1(o1, a1)

= ^

a1∈A1

_

a3∈A3

(g3(a3)⊗(ϕ2◦ϕ1)R(a3)(a1))→r1(o1, a1)

=↓1 (ϕ2◦ϕ1)R+(g3) (o1)

⋆=

↓1((ϕ2◦ϕ1)R+(g3)),↑1↓1((ϕ2◦ϕ1)R+(g3))

=(ϕ2◦ϕ1)_∧(hf3, g3i)

⊓

⊔ Proposition 1. Γ is a functor from L-ChuCorrstoL-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:

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 everyD∈ D there is some C∈ C such that F(C)∼=D.

Let us recall the definition of the notions required by the previous theorem:

Definition 14.

1. A functor F :C → D is faithful if for all objects A, B of a categoryC, the mapFA,B: HomC(A, B)→HomD(F(A), F(B))is injective.

2. Similarly,F is full ifFA,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

ΓC1,C2 :L-ChuCors(C1, C2)→L-CLLOS(Γ(C1), Γ(C2))

for any twoL-contextsC1andC2. 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 arrowhs, zifrom the set L-CLLOS(Γ(C1), Γ(C2)) there exists anL-Chu correspondenceϕ^hs,zi from the set L-ChuCors(C1, C2), for any two L-contexts Ci =hBi, Ai, rii fori={1,2}.

Let us define the following mappings:

– ϕ^hs,zi_L (o1) =Ext s h↓1↑1(χo1),↑1(χo1)i – ϕ^hs,zi_R (a2) =Int z h↓2(χa2),↑2↓2(χa2)i

↑2 ϕ^hs,zi_L (o1)

(a2) = ^

o2∈B2

ϕ^hs,zi_L (o1)(o2)→r2(o2, a2)

= ^

o2∈B2

Ext(s(h↓1↑1(χo1),↑1(χo1)i))(o2)→ ↓2(χa2)(o2)

=s h↓1↑1(χo1),↑1(χo1)i

2h↓2(χa2),↑2↓2(χa2)i

=h↓1↑1(χo1),↑1(χo1)i 1z h↓2(χa2),↑2↓2(χa2)i

= ^

a1∈A1

Int(z(h↓2(χa2),↑2↓2(χa2)i))(a1)→ ↑1(χo1)(a1)

= ^

a1∈A1

ϕ^hs,zi_R (a2)(a1)→r1(o1, a1)

=↓1 ϕ^hs,zi_R (a2) (o1)

Soϕ^hs,zi∈L-ChuCors(C1, C2) andΓC1,C2 is surjective, henceΓ is full. ⊓⊔

Lemma 7. Γ is faithfull

Proof. Now the point is to prove the injectivity ofΓC1,C2.

Consider two L-Chu correspondences ϕ1, ϕ2 from L-ChuCors(C1, C2) such thatϕ16=ϕ2, and let us fix the pair (o1, a2)∈B1×A2, such that

↑2 ϕ1L(o1)

(a2) =↓1 ϕ1R(a2)

(o1)6=↑2 ϕ2L(o1)

(a2) =↓1 ϕ2R(a2) (o1) Let us assume that either ↓1 ϕ1R(a2)

(o1)>↑2 ϕ2L(o1)

(a2) or that both values fromLare incomparable, that is equivalent to the following:

↓1 ϕ1R(a2)

(o1)→ ↑2 ϕ2L(o1)

(a2)<1

Now consider the L-concept h↓1 (ϕ1R(a2)), ϕ1R(a2)iand let us compare its images under the mappingsϕ1∨ andϕ2∨.

↑2 ϕ2L+ ↓1(ϕ1R(a2)) (a2)

= ^

o2∈B2

ϕ2L+(↓1(ϕ1R(a2)))(o2)→r2(o2, a2)

= ^

o2∈B2

_

b1∈B1

ϕ2L(b1)(o2)⊗ ↓1(ϕ1R(a2))(b1)

→r2(o2, a2)

= ^

b1∈B1

↓1(ϕ1R(a2))(b1)→ ^

o2∈B2

ϕ2L(b1)(o2)→r2(o2, a2)

= ^

b1∈B1

↓1(ϕ1R(a2))(b1)→ ↑2(ϕ2L(b1))(a2)

≤↓1(ϕ1R(a2))(o1)→ ↑2(ϕ2L(o1))(a2)

<1 because of the restriction given above Similarly, we can obtain:

↑2(ϕ1L+(↓1(ϕ1R(a2))))(a2) =

= ^

b1∈B1

↓1(ϕ1R(a2))(b1)→ ↑2(ϕ1L(b1))(a2)

= ^

b1∈B1

↓1(ϕ1R(a2))(b1)→ ↓1(ϕ1R(a2))(b1)

= 1

It means thatϕ1∨ h↓1(ϕ1R(a2)), ϕ1R(a2)i

6=ϕ2∨ h↓1(ϕ1R(a2)), ϕ1R(a2)i Hence ϕ1∨(h↓1 (ϕ1R(a2)), ϕ1R(a2)i) 6= ϕ2∨(h↓1 (ϕ1R(a2)), ϕ1R(a2)i) and ϕ1∨6=ϕ2∨. SoΓC1,C2 is injective andΓ is faithfull. ⊓⊔ Proposition 2. The functor Γ is an equivalence functor between L-ChuCors andL-CLLOS.

Proof. Fullness and faithfulness ofΓ is given by previous lemmas. Essential sur- jectivity on objects is ensured by the fact that given any object hhV,≈i,i of L-CLLOS there exists anL-contexthV, V,i, such thatΓ hV, V,i∼=hhV,≈i,i.

Hence, we can state thatΓ is the functor of equivalence betweenL-ChuCors and

L-CLLOS. ⊓⊔

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