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´ 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 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 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 ∈ L B × A L is a complete residuated lattice Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 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). Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 1 a b c d 0 � x ⊗ y = x ∧ y and x → y = { z ∈ L : x ⊗ z ≤ y } C 1 a 11 a 12 a 13 a 14 C 2 a 21 a 22 a 23 o 11 0 a c 1 o 21 0 a 1 o 12 a c 0 c o 22 c a c o 13 1 0 c 1 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 Definition L -fuzzy context C = � B , A , r � . Define mappings ↑ : L B → L A ; ↓ : L A → L B � � ; a ∈ A , f ∈ L B ↑ ( f )( a ) = � f ( o ) → r ( o , a ) o ∈ B � � ; o ∈ B , g ∈ L A ↓ ( g )( o ) = � g ( a ) → r ( o , a ) a ∈ A Definition An L -fuzzy concept is a pair � f , g � ∈ L B × L A such that ↑ ( f ) = g and ↓ ( g ) = f . f - Extent ( f is closed in C ) g - Intent ( g is closed in C ) Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 -fuzzy concept lattice 111 000c 110 1c1 011 0c0c 0001 a00c 010 1c0 101 0a1 ac0c 0a01 00c1 100c 100 0c1 0ac1 1001 ac0 0101 0a0 c0c 001 1c0c 0011 10c1 c00 00c 0c0 1101 0111 1011 C 1 a 11 a 12 a 13 a 14 0 a c 1 o 11 000 1111 a c 0 c o 12 1 0 c 1 o 13 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 -fuzzy concept lattice 11 0ac 01 1c aa cac 0a1 01c 0a ac c1c 011 C 2 a 21 a 22 a 23 0c 111 0 a 1 o 21 c a c o 22 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 -multifunction Definition An L - multifunction ϕ from X to Y is a mapping ϕ : X → L Y The transposition of ϕ : X → L Y is an L -multifunction ϕ t : Y → L X defined by ϕ t ( y )( x ) = ϕ ( x )( y ) The relation of ϕ : X → L Y is an L -fuzzy relation ϕ r ∈ L X × Y defined by ϕ r ( x , y ) = ϕ ( x )( y ) An ordering ϕ 1 ≤ ϕ 2 as ϕ 1 ( x )( y ) ≤ ϕ 2 ( x )( y ) for all x ∈ X and y ∈ Y . ϕ ∗ : L X → L Y , f ∈ L X � ϕ ∗ ( f )( y ) = ( f ( x ) ⊗ ϕ ( x )( y )). x ∈ X Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 Definition ( L -Bond) Let C 1 = � B 1 , A 1 , r 1 � , C 2 = � B 2 , A 2 , r 2 � be two L -fuzzy formal contexts. An L -multifunction β : B 1 → L A 2 is called an L -Bond from C 1 to C 2 if: β ( o 1 ) ∈ L A 2 closed in C 2 for all o 1 ∈ B 1 β t ( a 2 ) ∈ L B 1 closed in C 1 for all a 2 ∈ A 2 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 between two contexts 0 a 1 o 21 c a c o 22 a 21 a 22 a 23 a 11 a 12 a 13 a 14 0 1 1 o 11 0 a c 1 c 1 c o 12 a c 0 c 1 1 1 o 13 1 0 c 1 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 -fuzzy concept lattices 111 000c 11 110 1c1 011 0c0c 0001 a00c 0ac 010 1c0 101 0a1 ac0c 0a01 00c1 100c 01 aa 1c 100 0c1 cac 01c 0a1 0ac1 1001 ac0 0a0 c0c 001 0101 1c0c 0011 10c1 0a ac c1c 011 0c0 c00 00c 1101 0111 1011 0c 000 111 1111 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 Complete lattice L - Bonds ( C 1 , C 2 ) Lemma Let C 1 , C 2 be L-fuzzy formal contexts, and let β i ∈ L- Bonds ( C 1 , C 2 ) for all i ∈ I, and consider the following operations 1 ( � i ∈ I β i )( o ) = � i ∈ I ( β i ( o )) for all o ∈ B 1 . 2 ( � i ∈ I β i )( o ) = ↑ 2 ↓ 2 ( � i ∈ I ( β i ( o ))) = ↑ 2 ( � i ∈ I ↓ 2 ( β i ( o ))) for all o ∈ B 1 . The previous operations provide L- Bonds ( C 1 , C 2 ) with a complete lattice structure. Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 Complete lattice L - Bonds ( C 1 , C 2 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 c 1 c 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 c 1 c c 1 c 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 c 1 c 0 1 c 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 c 0 1 1 Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 Definition Let C 1 = � B 1 , A 1 , R 1 � , C 2 = � B 2 , A 2 , R 2 � be classical formal contexts. Direct product of C 1 and C 2 C 1 ∆ C 2 = � B 1 × A 2 , A 1 × B 2 , ∆ � , where (( o 1 , a 2 ) , ( a 1 , o 2 )) ∈ ∆ ≡ ( o 1 , a 1 ) ∈ R 1 OR ( o 2 , a 2 ) ∈ R 2 . Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego Saf´ arik, Koˇ sice, Slovakia and Univ. M´ alaga, Spain L -Bonds vs extents of direct product of two L -fuzzy contexts
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 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 . Univ. P. J. ˇ Ondrej Kr´ ıdlo, Stano Krajˇ ci, Manuel Ojeda-Aciego 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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