L-Chu Correspondences and Galois Functor

Ondrej Krıdlo and Stanislav Krajci

Abstract— This paper provides a categorical view on L-fuzzyconcept lattices inspired by paper [8]. Hideo Mori proved thata category of Chu correspondences between classical formalcontexts called ChuCors is equal to a category of join preservingmappings between complete lattices called Slat . Mori provedthe equality by introducing a Galois functor from ChuCors toSlat , that is an equivalence functor. We show that the Galoisfunctor for L-fuzzy extension of the category ChuCors is notthe equivalence functor to Slat .

I. INTRODUCTION

FORMAL CONCEPT ANALYSIS (FCA) introduced byGanter and Wille [5] as applied Lattice theory [4],

dealing with conceptual and hierarchical structures providedby object–attribute tables. L-fuzzy extension of basic notionsof FCA introduced by Belohlavek [1], [2] by substitutingclassical relation between objects-attribute sets by L-fuzzybinary relation, i.e. by mapping from set of all object-attribute pairs into a structure of truth values, completeresiduated lattice.

II. PRELIMINARIES

Definition 1: An algebra 〈L,∧,∨,⊗,→, 0, 1〉 is said to bea complete residuated lattice if

1) 〈L,∧,∨, 0, 1〉 is a complete bounded lattice with theleast element 0 and the greatest element 1,

2) 〈L,⊗, 1〉 is a commutative monoid,3) ⊗ and → are adjoint, i.e. a ⊗ b ≤ c if and only if

a ≤ b→ c, for all a, b, c ∈ L, where ≤ is the orderingin the lattice generated from ∧ and ∨.

We will use the notation Y X to refer to the set of mappingsfrom X to Y .

Let L be a complete residuated lattice, an L-fuzzy contextis a triple 〈B,A, r〉 consisting of a set of objects B, a set ofattributes A and an L-fuzzy binary relation r, i.e. a mappingr : B×A→ L, which can be alternatively understood as anL-fuzzy subset of B ×A

Consider an L-fuzzy context 〈B,A, r〉. A pair of mappings↑ : LB → LA and ↓ : LA → LB can be defined for everyf ∈ LB and g ∈ LA as follows:

↑ f(a) =∧o∈B

(f(o)→ r(o, a)

)(1)

↓ g(o) =∧a∈A

(g(a)→ r(o, a)

)(2)

Lemma 1: Let L be a complete residuated lattice, let r ∈LB×A be an L-fuzzy relation between B and A. Then the

Ondrej Krıdlo and Stanislav Krajci are with the Institute of ComputerScience, The University of P. J. Safarik of Kosice, Slovakia (email:o.kridlo@gmail.com, stanislav.krajci@upjs.sk).

This work was supported by a VEGA grant 1/0131/09.

pair of operators ↑ and ↓ form a Galois connection between〈LB ;⊆〉 and 〈LA;⊆〉, that is, ↑ : LB → LA and ↓ : LA →LB are antitonic and, furthermore, for all f ∈ LB and g ∈LA we have f ⊆ ↓↑f and g ⊆ ↑↓g.

Definition 2: Consider an L-fuzzy context C = 〈B,A, r〉.An L-fuzzy set of objects f ∈ LB (resp. an L-fuzzy set ofattributes g ∈ LA) is said to be closed in C iff f =↓↑ f(resp. g =↑↓ g).

Lemma 2: Under the conditions of Lemma 1, the fol-lowing equalities hold for arbitrary f ∈ LB and g ∈ LA,↑ f =↑↓↑ f and ↓ g =↓↑↓ g, that is, both ↓↑ f and ↑↓ g areclosed in C.

Definition 3: An L-fuzzy concept is a pair 〈f, g〉 suchthat ↑f = g, ↓g = f . The first component f is said to bethe extent of the concept, whereas the second component gis the intent of the concept.

The set of all L-fuzzy concepts associated to a fuzzycontext (B,A, r) will be denoted as CLL(B,A, r).

An ordering between L-fuzzy concepts is defined as fol-lows: 〈f1, g1〉 ≤ 〈f2, g2〉 if and only if f1 ⊆ f2 if and onlyif g1 ⊇ g2.

Theorem 1: The poset (CLL(B,A, r),≤) is a completelattice where∧

j∈J

〈fj , gj〉 =⟨ ∧

j∈J

fj , ↑( ∧

j∈J

fj)⟩

∨j∈J

〈fj , gj〉 =⟨↓( ∧

j∈J

gj),∧j∈J

gj

⟩.

III. L-CHU CORRESPONDENCES BETWEEN L-CONTEXTS

A notion of Chu correspondence introduced by Mori [8]as a pair of multifunctions between object sets of two formalcontexts and attribute sets in opposite way. The L-fuzzyextension of all basic notions of Chu correspondences isdescribed in [12].

Definition 4: Consider two L-fuzzy contexts Ci =〈Bi, Ai, ri〉, (i = 1, 2), then the pair ϕ = (ϕl, ϕr) is calleda correspondence from C1 to C2 if ϕl and ϕr are L-multifunctions, respectively, from B1 to B2 and from A2

to A1 (that is, ϕl : B1 → LB2 and ϕr : A2 → LA1 ).

459

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

↓1 (ϕr(a2))(o1) =↑2 (ϕl(o1))(a2) (3)

holds for all o1 ∈ B1 and a2 ∈ A2. A weak Chu correspon-dence ϕ is an L-Chu correspondence if ϕl(o1) is closed inC2 and ϕr(a2) is closed in C1 for all o1 ∈ B1 and a2 ∈ A2.We will denote the set of all Chu correspondences from C1

to C2 by L-ChuCors(C1, C2).

IV. CONNECTIONS OF RIGHT AND LEFT SIDES OF L-CHUCORRESPONDENCES

Now some technical properties of left and right side ofany L-Chu correspondence, which will be very helpful fordefining a category of L-Chu correspondences between L-fuzzy formal contexts.

Definition 5: Consider a mapping $ : X → LY . Letsdefine new mappings $∗ : LX → LY and $∗ : LY → LX

for all f ∈ LX and g ∈ LY put1) $∗(f)(y) =

∨x∈X(f(x)⊗$(x)(y))

2) $∗(g)(x) =∧

y∈Y $(x)(y)→ g(y)

Lemma 3: Let Ci = 〈Bi, Ai, ri〉 for i = 1, 2 be L-fuzzycontexts. Let ϕ = (ϕl, ϕr) ∈ L-ChuCors (C1, C2). Then forall f ∈ LB1 and g ∈ LA2 holds

↑2 (ϕl∗(f)) = ϕ∗r(↑1 (f)) (4)

↓1 (ϕr∗(g)) = ϕ∗l (↓2 (g)) (5)Proof: Let a2 ∈ A2

↑2 (ϕl∗(f))(a2)

=∧

o2∈B2

( ∨o1∈B1

(f(o1)⊗ ϕl(o1)(o2)

)→ r2(o2, a2)

)((∨i∈I

ai)→ a =∧i∈I

(ai → a))

=∧

o2∈B2

∧o1∈B1

((f(o1)⊗ ϕl(o1)(o2)

)→ r2(o2, a2)

)((a⊗ b)→ c = (a→ (b→ c)))

=∧

o2∈B2

∧o1∈B1

(f(o1)→

(ϕl(o1)(o2)→ r2(o2, a2)

))=

∧o1∈B1

(f(o1)→

∧o2∈B2

(ϕl(o1)(o2)→ r2(o2, a2)

))=

∧o1∈B1

(f(o1)→↑2 (ϕl(o1))(a2)

)(by definition of L-Chu correspondence)

=∧

o1∈B1

(f(o1)→↓1 (ϕr(a2))(o1)

)=

∧o1∈B1

(f(o1)→

∧a1∈A1

(ϕr(a2)(a1)→ r1(o1, a1)

))=

∧o1∈B1

∧a1∈A1

(f(o1)→

(ϕr(a2)(a1)→ r1(o1, a1)

))

=∧

o1∈B1

∧a1∈A1

((f(o1)⊗ ϕr(a2)(a1)

)→ r1(o1, a1)

)=

∧o1∈B1

∧a1∈A1

((ϕr(a2)(a1)⊗ f(o1)

)→ r1(o1, a1)

)=

∧o1∈B1

∧a1∈A1

(ϕr(a2)(a1)→

(f(o1)→ r1(o1, a1)

))=

∧a1∈A1

(ϕr(a2)(a1)→

∧o1∈B1

(f(o1)→ r1(o1, a1)

))=

∧a1∈A1

(ϕr(a2)(a1)→↑1 (f)(a1)

)= ϕ∗r(↑1 (f)))(a2)

Second equation can be proved symmetrically.

Corollary 1: Let Ci = 〈Bi, Ai, ri〉 for i = 1, 2 be L-contexts. If ϕ = (ϕl, ϕr) ∈ L-ChuCors (C1, C2), then forall o1 ∈ B1 and a2 ∈ A2 holds

ϕl(o1) =↓2 (ϕ∗r(↑1 (χo1)))

and

ϕr(a2) =↑1 (ϕ∗l (↓2 (χa2))).Proof: By definition of (.)∗ operator holds ϕl(o1) =

ϕl∗(χo1). By lemma 3 ↑2 (ϕl(o1)) = ϕ∗r(↑1 (χo1)). Bydefinition of Chu correspondences ϕl(o1) is closed in C2.ϕl(o1) =↓2 (↑2 (ϕl(o1))) =↓2 (ϕ∗r(↑1 (χo1))).

Second equation can be proved similarly.Lemma 4: Let Ci = 〈Bi, Ai, ri〉 be an L-context and ϕ ∈

L-ChuCors (C1, C2). Then

↑2 (ϕl∗(↓1↑1 (∨i∈I

fi)))(a2) =↑2 (∨i∈I

ϕl∗(fi))(a2) (6)

↓1 (ϕr∗(↑2↓2 (∨i∈I

gi)))(o1) =↓1 (∨i∈I

ϕr∗(gi))(o1) (7)

Proof:

↑2 (ϕl∗(↓1↑1 (∨i∈I

fi)))(a2)

=↑2 (ϕl∗(↓1 (∧i∈I

↑1 (fi))))(a2)

by lemma 3

= ϕ∗r(↑1↓1 (∧i∈I

↑1 (fi)))(a2)∧i∈I

↑1 (fi) is closed in C1

= ϕ∗r(∧i∈I

↑1 (fi))(a2)

=∧

a1∈A1

(ϕr(a2)(a1)→∧i∈I

↑1 (fi)(a1))

=∧i∈I

∧a1∈A1

(ϕr(a2)(a1)→↑1 (fi)(a1))

460

=∧i∈I

ϕ∗r(↑1 (fi))(a2)

=∧i∈I

↑2 (ϕl∗(fi))(a2)

=↑2 (∨i∈I

ϕl∗(fi))(a2)

V. CATEGORY L-ChuCors

Now a category of L-Chu correspondences between L-fuzzy formal contexts will be showed.• objects L-fuzzy formal contexts• arrows L-Chu correspondences• identity arrow ι : C → C of L-context C = 〈B,A, r〉

– ιl(o) =↓↑ (χo), for all o ∈ B– ιr(a) =↑↓ (χa), for all a ∈ A

• composition ϕ2 ◦ ϕ1 : C1 → C3 of arrowsϕ1 : C1 → C2, ϕ2 : C2 → C3, where Ci = 〈Bi, Ai, ri〉be L-fuzzy formal contexts for i ∈ {1, 2}

– (ϕ2◦ϕ1)l : B1 → LB3 and (ϕ2◦ϕ1)r : A3 → LA1

– (ϕ2 ◦ ϕ1)l(o1) =↓3↑3 (ϕ2l∗(ϕ1l(o1)))

– (ϕ2 ◦ ϕ1)r(a3) =↑1↓1 (ϕ1r∗(ϕ2r(a3)))

where

ϕ2l∗(ϕ1l(o1))(o3) =∨

o2∈B2

ϕ1l(o1)(o2)⊗ ϕ2l(o2)(o3)

and

ϕ1r∗(ϕ2r(a3))(a1) =∨

a2∈A2

ϕ2r(a3)(a2)⊗ϕ1r(a2)(a1)

• associativity of compositionϕ3 ◦ (ϕ2 ◦ ϕ1) = (ϕ3 ◦ ϕ2) ◦ ϕ1 for all ϕ1, ϕ2, ϕ3 ∈ L-ChuCors such that ϕi : Ci → Ci+1 for i ∈ {1, 2, 3},where C1, C2, C3, C4 are L-fuzzy contexts.

Proposition 1: Let C = 〈B,A, r〉 be the L-fuzzy formalcontext. The identity arrow ι : C → C defined above is anL-Chu correspondence from C to C.

Proof:

↑ (ιl(o))(a)

=∧b∈B

(↓↑ (χo)(b)→ r(b, a)) =↑↓↑ (χo)(a) =↑ (χo)(a)

=∧b∈B

(χo(b)→ r(b, a)) = r(o, a)

=∧d∈A

(χa(d)→ r(o, d)) =↓ (χa)(o)

=↓↑↓ (χa)(o) =∧d∈A

(↑↓ (χa)(d)→ r(o, d))

=↓ (ιr(a))(o)

Hence ι is a weak chu correspondence, but ιl(o) and ιr(a)are closed in C for all o ∈ B and all a ∈ A, so ι ∈L-ChuCors (C,C).

Proposition 2: Let Ci = 〈B,A, r〉, i ∈ {1, 2}be the L-fuzzy formal contexts. For identity arrowsof L-ChuCors ιi : Ci → Ci, i ∈ {1, 2} and ϕ ∈L-ChuCors (C1, C2) holds: (ι2 ◦ ϕ) = ϕ and (ϕ ◦ ι1) = ϕ.

Proof: (ι2 ◦ ϕ)l(o2) =↓2↑2 (ι2l∗(ϕl(o1)))(o2)

↑2 (ι2l∗(ϕl(o1)))(a2)

=∧

o2∈B2

(ι2l∗(ϕl(o1))(o2)→ r2(o2, a2))

=∧

o2∈B2

(∨

b2∈B2

(ι2l(b2)(o2)⊗ ϕl(o1)(b2))→ r2(o2, a2))

=∧

o2∈B2

∧b2∈B2

((ι2l(b2)(o2)⊗ ϕl(o1)(b2))→ r2(o2, a2))

=∧

o2∈B2

∧b2∈B2

(ϕl(o1)(b2)→ (ι2l(b2)(o2)→ r2(o2, a2)))

=∧

b2∈B2

(ϕl(o1)(b2)→∧

o2∈B2

(ι2l(b2)(o2)→ r2(o2, a2)))

=∧

b2∈B2

(ϕl(o1)(b2)→↑2 (ι2l(b2))(a2))

=∧

b2∈B2

(ϕl(o1)(b2)→↑2↓2↑2 (χ{b2})(a2))

=∧

b2∈B2

(ϕl(o1)(b2)→↑2 (χ{b2})(a2))

=∧

b2∈B2

(ϕl(o1)(b2)→ r2(b2, a2))

=↑2 (ϕl(o1))(a2)

So (ι2 ◦ ϕ)l(o2) =↓2↑2 (ϕl(o1)) = ϕl(o1). Second equationcan be proved similarly.

Proposition 3: Let Ci = 〈Bi, Ai, ri〉 be an L-fuzzy for-mal context for all i ∈ {1, 2, 3}. Let ϕj : Cj → Cj+1

be an L-Chu correspondence for all j ∈ {1, 2}, thencomposition ϕ2 ◦ ϕ1 : C1 → C3 defined above is an L-Chucorrespondence.

Proof: Let o1 ∈ B1 be an arbitrary object of C1 anda3 ∈ A3 be an arbitrary attribute of C3

↓1 ((ϕ2 ◦ ϕ1)r(a3))(o1)=↓1 (↑1↓1 (ϕ1r∗(ϕ2r(a3))))(o1)

=∧

a1∈A1

(↑1↓1 (ϕ1r∗(ϕ2r(a3)))(a1)→ r1(o1, a1))

=↓1↑1↓1 (ϕ1r∗(ϕ2r(a3)))(o1)=↓1 (ϕ1r∗(ϕ2r(a3)))(o1)

=∧

a1∈A1

(ϕ1r∗(ϕ2r(a3))(a1)→ r1(o1, a1))

=∧

a1∈A1

(∨

a2∈A2

(ϕ2r(a3)(a2)⊗ ϕ1r(a2)(a1))→ r1(o1, a1))

461

=∧

a1∈A1

∧a2∈A2

((ϕ2r(a3)(a2)⊗ ϕ1r(a2)(a1))→ r1(o1, a1))

=∧

a1∈A1

∧a2∈A2

(ϕ2r(a3)(a2)→ (ϕ1r(a2)(a1)→ r1(o1, a1)))

=∧

a2∈A2

(ϕ2r(a3)(a2)→∧

a1∈A1

(ϕ1r(a2)(a1)→ r1(o1, a1)))

=∧

a2∈A2

(ϕ2r(a3)(a2)→↓1 (ϕ1r(a2))(o1))

=∧

a2∈A2

(ϕ2r(a3)(a2)→↑2 (ϕ1l(o1))(a2))

=∧

a2∈A2

(ϕ2r(a3)(a2)→∧

o2∈B2

(ϕ1l(o1)(o2)→ r2(o2, a2)))

=∧

a2∈A2

∧o2∈B2

(ϕ2r(a3)(a2)→ (ϕ1l(o1)(o2)→ r2(o2, a2)))

=∧

a2∈A2

∧o2∈B2

((ϕ2r(a3)(a2)⊗ ϕ1l(o1)(o2))→ r2(o2, a2))

=∧

a2∈A2

∧o2∈B2

((ϕ1l(o1)(o2)⊗ ϕ2r(a3)(a2))→ r2(o2, a2))

=∧

a2∈A2

∧o2∈B2

(ϕ1l(o1)(o2)→ (ϕ2r(a3)(a2)→ r2(o2, a2)))

=∧

o2∈B2

(ϕ1l(o1)(o2)→∧

a2∈A2

(ϕ2r(a3)(a2)→ r2(o2, a2)))

=∧

o2∈B2

(ϕ1l(o1)(o2)→↓2 (ϕ2r(a3))(o2))

=∧

o2∈B2

(ϕ1l(o1)(o2)→↑3 (ϕ2l(o2))(a3))

=∧

o2∈B2

(ϕ1l(o1)(o2)→∧

o3∈B3

(ϕ2l(o2)(o3)→ r3(o3, a3)))

=∧

o2∈B2

∧o3∈B3

(ϕ1l(o1)(o2)→ (ϕ2l(o2)(o3)→ r3(o3, a3)))

=∧

o2∈B2

∧o3∈B3

((ϕ1l(o1)(o2)⊗ ϕ2l(o2)(o3))→ r3(o3, a3))

=∧

o3∈B3

(∨

o2∈B2

(ϕ1l(o1)(o2)⊗ ϕ2l(o2)(o3))→ r3(o3, a3))

=∧

o3∈B3

(ϕ2l∗(ϕ1l(o1))(o3)→ r3(o3, a3))

=↑3 (ϕ2l∗(ϕ1l(o1)))(a3)=↑3↓3↑3 (ϕ2l∗(ϕ1l(o1)))(a3)

=∧

o3∈B3

(↓3↑3 (ϕ2l∗(ϕ1l(o1)))(o3)→ r3(o3, a3))

=↑3 ((ϕ2 ◦ ϕ1)l(o1))(a3)

Hence the composition is weak L-Chu correspondence. Butleft or right side is closed set of objects in C3 or closed set ofattributes in C1 respectively. So the composition is an L-Chucorrespondence.

Proposition 4: Let Ci = 〈Bi, Ai, ri〉 for i ∈ {1, 2, 3, 4}be the L-fuzzy contexts. ϕi : Ci → Ci+1 for i ∈ {1, 2, 3}be the Chu correspondences. Then the associativity of com-

position holds ϕ3 ◦ (ϕ2 ◦ ϕ1) = (ϕ3 ◦ ϕ2) ◦ ϕ1.Proof:

((ϕ3 ◦ ϕ2) ◦ ϕ1)l(o1)=↓4↑4 ((ϕ3 ◦ ϕ2)l∗(ϕ1l(o1)))=↓4↑4 (↓4↑4 (ϕ3l∗(ϕ2l∗(ϕ1l(o1)))))by the property of closure operator=↓4↑4 (ϕ3l∗(ϕ2l∗(ϕ1l(o1))))by lemma 3=↓4↑4 (ϕ3l∗(↓3↑3 (ϕ2l∗(ϕ1l(o1)))))=↓4↑4 (ϕ3l∗((ϕ2 ◦ ϕ1)l(o1))) == (ϕ3 ◦ (ϕ2 ◦ ϕ1))l(o1)

VI. GALOIS FUNCTOR

Lets define a functor that assigns to every L-formal contextC a complete lattice Gal(C), the concept lattice of C andto every L-Chu correspondence assigns a join preservingmapping between concept lattices of contexts.

Definition 6: Let Ci = 〈Bi, Ai, ri〉 be an L-formal con-text for i ∈ {1, 2}, ϕ ∈ L-ChuCors (C1, C2) and 〈f, ↑1 (f)〉,〈↓2 (g), g〉 be L-concepts from Gal(C1) or Gal(C2) re-spectively. Define ϕ< : Gal(C1) → Gal(C2) and ϕ> :Gal(C2)→ Gal(C1) by

ϕ<(〈f, ↑1 (f)〉) = 〈↓2↑2 (ϕl∗(f)), ↑2 (ϕl∗(f))〉 (8)

ϕ>(〈↓2 (g), g〉) = 〈↓1 (ϕr∗(g)), ↑1↓1 (ϕr∗(g))〉. (9)

Lemma 5: Let Ci = 〈Bi, Ai, ri〉 be an L-fuzzy formalcontext for i ∈ {1, 2} and ϕ ∈ L-ChuCors (C1, C2).Assign b ∈ LB1×A2 as a new L-relation defined by anL-bond βϕ([12],[8]), namely, b(o1, a2) =↑2 (ϕl(o1))(a2),for all o1 ∈ B1 and a2 ∈ A2. ↑b, ↓b are up- and down-arrowmappings defined on relation b. For all f ∈ LB1 ,g ∈ LA2 holds

↑2 (ϕl∗(f)) =↑b (f) and ↓1 (ϕr∗(g)) =↓b (g).Proof:

↓1 (ϕr∗(g))

=∧

a1∈A1

(ϕr∗(g)→ r1(o1, a1))

=∧

a1∈A1

(∨

a2∈A2

(ϕr(a2)(a1)⊗ g(a2)))→ r1(o1, a1))

=∧

a1∈A1

∧a2∈A2

((ϕr(a2)(a1)⊗ g(a2))→ r1(o1, a1))

=∧

a1∈A1

∧a2∈A2

((g(a2)⊗ ϕr(a2)(a1))→ r1(o1, a1))

=∧

a1∈A1

∧a2∈A2

(g(a2)→ (ϕr(a2)(a1)→ r1(o1, a1)))

462

=∧

a2∈A2

(g(a2)→∧

a1∈A1

(ϕr(a2)(a1)→ r1(o1, a1)))

=∧

a2∈A2

(g(a2)→↓1 (ϕr(a2))(o1))

=∧

a2∈A2

(g(a2)→ βϕ(o1)(a2))

=∧

a2∈A2

(g(a2)→ b(o1, a2))

=↓b (g)(o1)

The second equation can be proved similarly.

Lemma 6: For all f ∈ LB1 and g ∈ LA2 holds

f ≤↓1 (ϕr∗(g))⇔ g ≤↑2 (ϕl∗(f)).Proof: ⇐

↓1 (ϕr∗(g))=↓b (g)

=∧

a2∈A2

(g(a2)→ b(o1, a2))

now precondition will be used

≥∧

a2∈A2

(↑2 (ϕl∗(f))(a2)→ b(o1, a2))

from lemma 5

=∧

a2∈A2

(↑b (f)(a2)→ b(o1, a2))

=↓b↑b (f)(o1) ≥ f(o1)

⇒ Similar.

Proposition 5: Pair of mappings 〈ϕ<, ϕ>〉 for any ϕ ∈L-ChuCors (C1, C2), where Ci = 〈Bi, Ai, ri〉 be L-fuzzyformal context for i ∈ {1, 2}, forms a Galois connectionbetween concept lattices of C1 and C2, namely, for all f ∈LB1 closed in C1 and g ∈ LA2 closed in C2 holds

〈f, ↑1 (f)〉 ≤ 〈↓1 (ϕr∗(g)), ↑1↓1 (ϕr∗(g))〉 ⇔

⇔ 〈↓2↑2 (ϕl∗(f)), ↑2 (ϕl∗(f))〉 ≤ 〈↓2 (g), g〉

.Proof: Equivalence above is equal to equivalence

f ≤↓1 (ϕr∗(g))⇔ g ≤↑2 (ϕl∗(f)),

which holds from lemma 6.

Lemma 7: Let Ci = 〈Bi, Ai, ri〉 be an L-context for i ∈{1, 2}. For every ϕ ∈ L-ChuCors (C1, C2) holds

1) ϕ< is supremum-preserving,2) ϕ> is infimum-preserving.

Proof:1

From lemma 4

ϕ<(〈↓1↑1 (∨i∈I

fi),∧i∈I

↑1 (fi)〉)

= 〈↓2↑2 (ϕl∗(↓1↑1 (∨i∈I

fi))), ↑2 (ϕl∗(↓1↑1 (∨i∈I

fi)))〉

= 〈↓2↑2 (∨i∈I

ϕl∗(fi))(a2),∧i∈I

↑2 (ϕl∗(fi))(a2)〉

Hence to a join of L-concepts

〈↓1↑1 (∨i∈I

fi),∧i∈I

↑1 (fi)〉

from CLL(C1), ϕ< assigns a join of L-concepts

〈↓2↑2 (∨i∈I

ϕl∗(fi))(a2),∧i∈I

↑2 (ϕl∗(fi))(a2)〉

from CLL(C2).2 Similarly with lemma 4 it can be proved that to a

meet of L-concepts 〈∧

i∈I ↓2 (gi), ↑2↓2 (∨

i∈I gi)〉 fromCLL(C2) ϕ> assigns a meet of L-concepts 〈

∧i∈I ↓1

(ϕr∗(gi)), ↑1↓1 (∨

i∈I gi)〉 from CLL(C1).

Lemma 8: Let C = 〈B,A, r〉 be an L-context. For ι ∈L-ChuCors (C,C), ι : C → C, an identity arrow of categoryL-ChuCors holds for all f ∈ LB and all g ∈ LA

↑ (ιl∗(f)) =↑ (f) and ↓ (ιr∗(g)) =↓ (g).Proof: Omited due to lack of space.

Definition 7: Define a mapping Gal, which to every L-context C assigns a complete lattice of L-concepts of C, andto every L-Cu correspondence assigns asigns a supremum-preserving mapping ϕ<.

Theorem 2: Gal is a functor from category L-ChuCorsto category Slat .

Proof: Let C be an arbitrary L-context. Gal(C) is acomplete concept lattice 〈CLL(C),≤〉.

Let 〈f, ↑ (f)〉 ∈ CLL(C), then by lemma 8

ιC∗(〈f, ↑ (f)〉)= 〈↓↑ (ιl∗(f)), ↑ (ιl∗(f))〉= 〈↓↑ (f), ↑ (f)〉 = 〈f, ↑ (f)〉

Hence Gal assigns to an identity arrow of L-ChuCors anidentity arrow of Slat , i.e. Gal(ιC) = ιGal(C).

Let Ci = 〈Bi, Ai, ri〉 be an L-context for all i = {1, 2, 3},and for all j ∈ {1, 2} let ϕj ∈ L-ChuCors (Cj , Cj+1). Let

463

TABLE ITABLES WITH ORDER RELATION AND CONNECTIVES ⊗ AND→

≤ 1 a b c d 01 •a • •b • •c • • • •d • • •0 • • • • • •

⊗ 1 a b c d 01 1 a b c d 0a a a c c d 0b b c b c 0 0c c c c c 0 0d d d 0 0 d 00 0 0 0 0 0 0

→ 1 a b c d 01 1 a b c d 0a 1 1 b b 0 0b 1 a 1 a d dc 1 1 1 1 d dd 1 1 b b 1 b0 1 1 1 1 1 1

〈f, ↑ (f)〉 ∈ CLL(C1)

(ϕ2 ◦ ϕ1)<(〈f, ↑1 (f)〉)= 〈↓3↑3 ((ϕ2 ◦ ϕ1)l∗(f)), ↑3 ((ϕ2 ◦ ϕ1)l∗(f))〉

from definitionof composition of arrows= 〈↓3↑3↓3↑3 (ϕ2l∗(ϕ1l∗(f))), ↑3↓3↑3 (ϕ2l∗(ϕ1l∗(f)))〉= 〈↓3↑3 (ϕ2l∗(ϕ1l∗(f))), ↑3 (ϕ2l∗(ϕ1l∗(f)))〉= 〈↓3↑3 (ϕ2l∗(↓2↑2 (ϕ1l∗(f)))), ↑3 (ϕ2l∗(↓2↑2 (ϕ1l∗(f))))〉= ϕ2∗(〈↓2↑2 (ϕ1l∗(f)), ↑2 (ϕ1l∗(f))〉)= ϕ2∗(ϕ1∗(〈f, ↑1 (f)〉))

Hence Gal(ϕ2◦ϕ1) = (ϕ2◦ϕ1)< = ϕ2∗◦ϕ1∗ = Gal(ϕ2)◦Gal(ϕ1)

So Gal : L-ChuCors → Slat is a functor.

VII. GALOIS FUNCTOR IS NOT THE EQUIVALENCEFUNCTOR BETWEEN L-ChuCors AND Slat

Mori in [8] showed that Gal is the bijection mapping fromChuCors(C1, C2) to Slat (Gal(C1), Gal(C2)), that inducethat functor Gal is full and faithfull.

Example 1: C1 = 〈{o1}, {a1}, r1(o1, a1) = c〉 and C2 =〈{o2}, {a2}, r2(o2, a2) = 0〉 be L-fuzzy formal contexts.Where L = 〈{1, a, b, c, d, 0},≤, 0, 1,⊗,→〉, where latticeordering and binary operations ⊗ and → are described intable I.

For all k, l,m ∈ L adjoint k ⊗ l ≤ m iff k ≤ l → mholds.

Notation 〈l〉 labels mapping from one element set to L,where l ∈ L.CLL(C1) = {〈〈1〉, 〈c〉〉, 〈〈a〉, 〈b〉〉, 〈〈b〉, 〈a〉〉, 〈〈c〉, 〈1〉〉}CLL(C2) = {〈〈1〉, 〈0〉〉, 〈〈b〉, 〈d〉〉, 〈〈d〉, 〈b〉〉, 〈〈0〉, 〈1〉〉}There are only two L-Chu correspondences from C1 to

C2

〈ϕl1(o1)(o2) = d;ϕr1(a2)(a1) = a〉

ϕl2(o1)(o2) = 0;ϕr2(a2)(a1) = c〉,

but count of join preserving mappings between conceptlattices Gal(C1) and Gal(C2) are more than two.

There doesn’t exist any surjectionγ : L-ChuCors (C1, C2)→ Slat (Gal(C1), Gal(C2)),so the functor Gal cannot be full.

Definition of fullness of functor [3] is that induced map-ping (like γ of functor Gal) is surjective for every hom set.From example we can see that there exist hom sets wheremapping γ cannot be surjective.

In [3] there is a theorem about relationship of equivalenceand fullness and faithfullness.

Theorem 3: Let F : C → D be an equivalence ofcategories and G : D → C is pseudo-inverse to F . ThenF and G are full and faithfull.

So if Gal is not full then it cannot be the equivalence functorbetween ChuCors and Slat .

VIII. CONCLUSION

We have provided the L-fuzzy extension of category ofChu correspondences between formal contexts. We havedefined the Galois functor from the category L-ChuCors tocategory Slat . It is showed that functor Gal is not full, hencecannot be the equivalence functor, despite of the fact that Galis the equivalence functor from classical ChuCors to Slat .

Mori in [8] showed that functor Gal because of equiva-lence preserves ∗−autonomity of category Slat . Our futurework will be to study this topics and looking for some otherinteresting functorial or categorical properties.

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