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:[email protected], [email protected]).
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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978-1-4244-6337-4/10/$26.00 @2010 IEEE
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
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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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