L -Bonds vs extents of direct product of two L -fuzzy contexts - - PowerPoint PPT Presentation

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L -Bonds vs extents of direct product of two L -fuzzy contexts - - PowerPoint PPT Presentation

Sep 23, 2022 348 likes •580 views

L -fuzzy FCA L - Bonds Direct product of two L -fuzzy contexts Relationship between L - Bonds ( C 1 , C 2 ) and C 1 C 2 L -Bonds vs extents of direct product of two L -fuzzy contexts Ondrej Kr dlo, Stano Kraj ci, Manuel Ojeda-Aciego

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-Bonds vs extents of direct product of two L-fuzzy contexts

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and

  • Univ. M´

alaga, Spain

CLA 2010, Sevilla, Spain

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition An L-fuzzy formal context is a triple B, A, r a set of objects B a set of attributes A an L-fuzzy binary relation r ∈ LB×A L is a complete residuated lattice

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition A complete residuated lattice is an algebra L = L, ∧, ∨, ⊗, →, 0, 1 where (a) L, ∧, ∨, 0, 1 is a bounded lattice with the least element 0 and the greatest element 1, (b) L, ⊗, 1 is a commutative monoid, (c) ⊗ and → satisfy adjointness, i.e. a ⊗ b ≤ c ⇐ ⇒ a ≤ b → c for each a, b, c ∈ L (≤ is the lattice ordering).

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

1 a b c d

x ⊗ y = x ∧ y and x → y =

  • {z ∈ L : x ⊗ z ≤ y}

C1 a11 a12 a13 a14

  • 11

a c 1

  • 12

a c c

  • 13

1 c 1 C2 a21 a22 a23

  • 21

a 1

  • 22

c a c

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition L-fuzzy context C = B, A, r. Define mappings ↑: LB → LA; ↓: LA → LB ↑ (f )(a) =

  • ∈B
  • f (o) → r(o, a)
  • ; a ∈ A, f ∈ LB

↓ (g)(o) =

a∈A

  • g(a) → r(o, a)
  • ; o ∈ B, g ∈ LA

Definition An L-fuzzy concept is a pair f , g ∈ LB × LA such that ↑ (f ) = g and ↓ (g) = f . f - Extent (f is closed in C) g - Intent (g is closed in C)

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-fuzzy concept lattice

111 000c 110 0c0c 1c1 0001 011 a00c 1c0 0a01 010 ac0c 101 00c1 0a1 100c 100 0ac1 ac0 0101 0a0 1c0c c0c 0011 0c1 1001 0c0 1101 c00 0111 00c 1011 000 1111 001 10c1

C1 a11 a12 a13 a14

  • 11

a c 1

  • 12

a c c

  • 13

1 c 1

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-fuzzy concept lattice

11 0ac 01 cac aa 01c 1c 0a1 0a c1c ac 011 0c 111

C2 a21 a22 a23

  • 21

a 1

  • 22

c a c

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-multifunction

Definition An L-multifunction ϕ from X to Y is a mapping ϕ: X → LY The transposition of ϕ: X → LY is an L-multifunction ϕt : Y → LX defined by ϕt(y)(x) = ϕ(x)(y) The relation of ϕ: X → LY is an L-fuzzy relation ϕr ∈ LX×Y defined by ϕr(x, y) = ϕ(x)(y) An ordering ϕ1 ≤ ϕ2 as ϕ1(x)(y) ≤ ϕ2(x)(y) for all x ∈ X and y ∈ Y . ϕ∗ : LX → LY ,f ∈ LX ϕ∗(f )(y) =

  • x∈X

(f (x) ⊗ ϕ(x)(y)).

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition (L-Bond) Let C1 = B1, A1, r1, C2 = B2, A2, r2 be two L-fuzzy formal

  • contexts. An L-multifunction β : B1 → LA2 is called an L-Bond

from C1 to C2 if: β(o1) ∈ LA2 closed in C2 for all o1 ∈ B1 βt(a2) ∈ LB1 closed in C1 for all a2 ∈ A2

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-Bonds between two contexts

a 1

  • 21

c a c

  • 22

a21 a22 a23 a11 a12 a13 a14 1 1

  • 11

a c 1 c 1 c

  • 12

a c c 1 1 1

  • 13

1 c 1

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

L-fuzzy concept lattices

111 000c 110 0c0c 1c1 0001 011 a00c 1c0 0a01 010 ac0c 101 00c1 0a1 100c 100 0ac1 ac0 0101 0a0 1c0c c0c 0011 0c1 1001 0c0 1101 c00 0111 00c 1011 000 1111 001 10c1

11 0ac 01 cac aa 01c 1c 0a1 0a c1c ac 011 0c 111

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Complete lattice L-Bonds(C1, C2)

Lemma Let C1, C2 be L-fuzzy formal contexts, and let βi ∈ L-Bonds(C1, C2) for all i ∈ I, and consider the following

  • perations

1 ( i∈I βi)(o) = i∈I(βi(o)) for all o ∈ B1. 2 ( i∈I βi)(o) =↑2↓2 ( i∈I(βi(o))) =↑2 ( i∈I ↓2 (βi(o))) for

all o ∈ B1. The previous operations provide L-Bonds(C1, C2) with a complete lattice structure.

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Complete lattice L-Bonds(C1, C2)

1 1 1 1 1 1 1 1 1 1 1 1 c 1 c 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c 1 c 1 1 1 1 1 1 c 1 c 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c 1 c 1 1 1 1 1 1 c 1 1 1 1 1 1 1 1 1 1 1 c 1 1

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition Let C1 = B1, A1, R1, C2 = B2, A2, R2 be classical formal

  • contexts. Direct product of C1 and C2

C1∆C2 = B1 × A2, A1 × B2, ∆, where ((o1, a2), (a1, o2)) ∈ ∆ ≡ (o1, a1) ∈ R1 OR (o2, a2) ∈ R2.

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition Let L be complete residuated lattice (structure of truth values). Lets define (Negation) Unary operator ¬ : L → L, such that ¬k = k → 0 for k ∈ L (Disjunction) Binary operation ⋉ : L × L → L, such that k ⋉ m = ¬k → m for all k, m ∈ L. In general: k ≤ ¬¬k ¬k = ¬¬¬k k → m ≤ ¬m → ¬k Definition Law of double negation:¬¬k = k for all k ∈ L.

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Definition Direct product of two L-contexts C1 = B1, A1, r1, C2 = B2, A2, r2 is an L-context C1∆C2 = B1 × A2, A1 × B2, ∆, such that ∆((o1, a2), (a1, o2)) = r1(o1, a1) ⋉ r2(o2, a2) = ¬r1(o1, a1) → r2(o2, a2). Definition (Complement of L-fuzzy context) Complement of L-fuzzy formal context B, A, r B, A, ¬r; ¬r(o, a) = r(o, a) → 0 Uparrow and Downarrow mappings ↑¬ and ↓¬ .

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Theorem Let Ci = Bi, Ai, ri be L-fuzzy contexts for i ∈ {1, 2}, where L satisfies the double negation law. Let β ∈ L-Mfn(B1, A2). Then:

1 If βr is an extent of C1∆C2, then β ∈ L-Bond(C1, C2). 2 If β ∈ L-Bond(C1, C2) and

β∗(↓¬1↑¬1 (χo1))(a2) =

  • a1∈A1

(↑¬1 (χo1)(a1) → ↑2↓2 (β∗(↓¬1 (χa1)))(a2)) then rβ is an extent of C1∆C2.

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Lemma C1 = B1, A1, r1, C2 = B2, A2, r2 be two L-contexts ϕ : B1 → LA2 and ψ : A1 → LB2 ↓2 (ϕ∗(↓¬1 (χa1)))(o2) =↑∆ (ϕr)(o2, a1) =↑1 (tϕ∗(↑¬2 (χo2)))(a1) and ↑2 (ψ∗(↑¬1 (χo1)))(a2) =↓∆ (ψr)(o1, a2) =↓1 (tψ∗(↓¬2 (χa2)))(o1)

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Let rβ be an extent of C1∆C2, then exists some ϕ ∈ L-Mfn(A1, B2), such that β(o1)(a2) =↓∆ (ϕr)(o1, a2) =↑2 (ϕ∗(↑¬1 (χo1)))(a2) =

tβ(a2)(o1) =↓∆ (ϕr)(o1, a2) =↓1 (tϕ∗(↓¬2 (χa2)))(o1)

β ∈ L-Bonds(C1, C2)

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

If β ∈ L-Bonds(C1, C2), then β(o1)(a2) = . . . · · · = β∗(↓¬1↑¬1 (χo1))(a2) =

  • a1∈A1

(↑¬1 (χo1)(a1) → ↑2↓2 (β∗(↓¬1 (χa1)))(a2)) = . . . · · · =↓∆↑∆ (βr)(o1, a2)

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts

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L-fuzzy FCA L-Bonds Direct product of two L-fuzzy contexts Relationship between L-Bonds(C1, C2) and C1∆C2

Thank you for your attention!

Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego

  • Univ. P. J. ˇ

Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L-Bonds vs extents of direct product of two L-fuzzy contexts