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Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and n -Alg Interrelation among F -Top Sys, F -Top and Frm Future Direction References On Categorical

  1. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Topological System A Topological system is a triple ( X , | = , A ) where X is a set, A is a frame and | =, is a relation | = ⊆ X × A , matches the logic of finite observations. Formally, Purbita Jana BLAST 2013 12 / 67

  2. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Topological System A Topological system is a triple ( X , | = , A ) where X is a set, A is a frame and | =, is a relation | = ⊆ X × A , matches the logic of finite observations. Formally, = � S ⇔ x | If S is a finite subset of A , then x | = a for all a ∈ S . Purbita Jana BLAST 2013 12 / 67

  3. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Topological System A Topological system is a triple ( X , | = , A ) where X is a set, A is a frame and | =, is a relation | = ⊆ X × A , matches the logic of finite observations. Formally, = � S ⇔ x | If S is a finite subset of A , then x | = a for all a ∈ S . = � S ⇔ x | If S is any subset of A , then x | = a for some a ∈ S . Purbita Jana BLAST 2013 12 / 67

  4. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Continuous map = ′ , B ) be topological systems. A Let D = ( X , | = , A ) and E = ( Y , | continuous map f : D − → E is a pair ( f 1 , f 2 ) where, Purbita Jana BLAST 2013 13 / 67

  5. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Continuous map = ′ , B ) be topological systems. A Let D = ( X , | = , A ) and E = ( Y , | continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. Purbita Jana BLAST 2013 13 / 67

  6. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Continuous map = ′ , B ) be topological systems. A Let D = ( X , | = , A ) and E = ( Y , | continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. f 2 : B − → A is a frame homomorphism and Purbita Jana BLAST 2013 13 / 67

  7. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Continuous map = ′ , B ) be topological systems. A Let D = ( X , | = , A ) and E = ( Y , | continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. f 2 : B − → A is a frame homomorphism and = ′ b , for all x ∈ X and b ∈ B . x | = f 2 ( x ) iff f 1 ( x ) | Purbita Jana BLAST 2013 13 / 67

  8. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Identity map Let D = ( X , | = , A ) be a topological system. The identity map I D : D − → D is a pair ( I 1 , I 2 ) defined by Purbita Jana BLAST 2013 14 / 67

  9. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Identity map Let D = ( X , | = , A ) be a topological system. The identity map I D : D − → D is a pair ( I 1 , I 2 ) defined by I 1 : X − → X I 2 : A − → A Purbita Jana BLAST 2013 14 / 67

  10. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let D = ( X , | Purbita Jana BLAST 2013 15 / 67

  11. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. Purbita Jana BLAST 2013 15 / 67

  12. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by Purbita Jana BLAST 2013 15 / 67

  13. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by g 1 ◦ f 1 : X − → Z f 2 ◦ g 2 : C − → A Purbita Jana BLAST 2013 15 / 67

  14. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Sys Top Sys Topological systems together with continuous maps form the category Top Sys. Purbita Jana BLAST 2013 16 / 67

  15. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Top Top Topological spaces together with continuous maps form the category Top. Purbita Jana BLAST 2013 17 / 67

  16. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Frm Frm Frames together with frame homomorphisms form the category Frm. Purbita Jana BLAST 2013 18 / 67

  17. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Ext where ext ( a ) = { x | x | = a } and ext ( A ) = { ext ( a ) } a ∈ A . Purbita Jana BLAST 2013 19 / 67

  18. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References J Purbita Jana BLAST 2013 20 / 67

  19. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References fm Purbita Jana BLAST 2013 21 / 67

  20. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References S where Fm ( A , 2) = { frame homomorphism : A − → 2 } and x | = ∗ a iff x ( a ) = ⊤ . Purbita Jana BLAST 2013 22 / 67

  21. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . Purbita Jana BLAST 2013 23 / 67

  22. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . Purbita Jana BLAST 2013 23 / 67

  23. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . 3 Ext ◦ S is the right adjoint to the functor fm ◦ J . Purbita Jana BLAST 2013 23 / 67

  24. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Fuzzy Topological System A fuzzy topological system is a triple ( X , | = , A ), where X is a non-empty set, A is a frame and | = is a [0 , 1]- fuzzy relation from X to A such that Purbita Jana BLAST 2013 24 / 67

  25. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Fuzzy Topological System A fuzzy topological system is a triple ( X , | = , A ), where X is a non-empty set, A is a frame and | = is a [0 , 1]- fuzzy relation from X to A such that if S is a finite subset of A , then = � S ) = inf { gr ( x | gr ( x | = s ) : s ∈ S } Purbita Jana BLAST 2013 24 / 67

  26. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Fuzzy Topological System A fuzzy topological system is a triple ( X , | = , A ), where X is a non-empty set, A is a frame and | = is a [0 , 1]- fuzzy relation from X to A such that if S is a finite subset of A , then = � S ) = inf { gr ( x | gr ( x | = s ) : s ∈ S } if S is any subset of A , then = � S ) = sup { gr ( x | gr ( x | = s ) : s ∈ S } Purbita Jana BLAST 2013 24 / 67

  27. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Continuous map = ′ , B ) be fuzzy topological Let D = ( X , | = , A ) and E = ( Y , | systems. A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, Purbita Jana BLAST 2013 25 / 67

  28. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Continuous map = ′ , B ) be fuzzy topological Let D = ( X , | = , A ) and E = ( Y , | systems. A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. Purbita Jana BLAST 2013 25 / 67

  29. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Continuous map = ′ , B ) be fuzzy topological Let D = ( X , | = , A ) and E = ( Y , | systems. A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. f 2 : B − → A is a frame homomorphism and Purbita Jana BLAST 2013 25 / 67

  30. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Continuous map = ′ , B ) be fuzzy topological Let D = ( X , | = , A ) and E = ( Y , | systems. A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, f 1 : X − → Y is a function. f 2 : B − → A is a frame homomorphism and = ′ b ), for all x ∈ X and b ∈ B . gr ( x | = f 2 ( b )) = gr ( f 1 ( x ) | Purbita Jana BLAST 2013 25 / 67

  31. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Identity map Let D = ( X , | = , A ) be a fuzzy topological system. The identity map I D : D − → D is a pair ( I 1 , I 2 ) defined by Purbita Jana BLAST 2013 26 / 67

  32. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Identity map Let D = ( X , | = , A ) be a fuzzy topological system. The identity map I D : D − → D is a pair ( I 1 , I 2 ) defined by I 1 : X − → X I 2 : A − → A Purbita Jana BLAST 2013 26 / 67

  33. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let D = ( X , | Purbita Jana BLAST 2013 27 / 67

  34. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be fuzzy continuous maps. Purbita Jana BLAST 2013 27 / 67

  35. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be fuzzy continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by Purbita Jana BLAST 2013 27 / 67

  36. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be fuzzy continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by g 1 ◦ f 1 : X − → Z f 2 ◦ g 2 : C − → A Purbita Jana BLAST 2013 27 / 67

  37. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Sys Fuzzy Top Sys Fuzzy topological systems together with continuous maps form the category Fuzzy Top Sys. Purbita Jana BLAST 2013 28 / 67

  38. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Fuzzy Top Top Fuzzy topological spaces together with fuzzy continuous maps form the category Fuzzy Top. Purbita Jana BLAST 2013 29 / 67

  39. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Frm Frm Frames together with frame homomorphisms form the category Frm. Purbita Jana BLAST 2013 30 / 67

  40. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Ext where ext ( a ) : X − → [0 , 1] s.t. ext ( a )( x ) = gr ( x | = a ) and ext ( A ) = { ext ( a ) } a ∈ A . Purbita Jana BLAST 2013 31 / 67

  41. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References J Purbita Jana BLAST 2013 32 / 67

  42. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References fm Purbita Jana BLAST 2013 33 / 67

  43. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References S where Hom ( A , [0 , 1]) = { frame homomorphism v : A − → [0 , 1] } and gr ( v | = ∗ a ) = v ( a ). Purbita Jana BLAST 2013 34 / 67

  44. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . Purbita Jana BLAST 2013 35 / 67

  45. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . Purbita Jana BLAST 2013 35 / 67

  46. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . 3 Ext ◦ S is the right adjoint to the functor fm ◦ J . Purbita Jana BLAST 2013 35 / 67

  47. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Definition of Fuzzy Topological System given by Apostolos and Paiva A fuzzy topological system is an object ( U , X , α ) of Dial I ( Set ) such that X is a frame and α satisfies the following conditions: Purbita Jana BLAST 2013 36 / 67

  48. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Definition of Fuzzy Topological System given by Apostolos and Paiva A fuzzy topological system is an object ( U , X , α ) of Dial I ( Set ) such that X is a frame and α satisfies the following conditions: If S is a finite subset of X , then α ( u , � S ) ≤ α ( u , x ) for all x ∈ S . Purbita Jana BLAST 2013 36 / 67

  49. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Definition of Fuzzy Topological System given by Apostolos and Paiva A fuzzy topological system is an object ( U , X , α ) of Dial I ( Set ) such that X is a frame and α satisfies the following conditions: If S is a finite subset of X , then α ( u , � S ) ≤ α ( u , x ) for all x ∈ S . If S is any subset of X , then α ( u , � S ) ≤ α ( u , x ) for some x ∈ S . Purbita Jana BLAST 2013 36 / 67

  50. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Definition of Fuzzy Topological System given by Apostolos and Paiva A fuzzy topological system is an object ( U , X , α ) of Dial I ( Set ) such that X is a frame and α satisfies the following conditions: If S is a finite subset of X , then α ( u , � S ) ≤ α ( u , x ) for all x ∈ S . If S is any subset of X , then α ( u , � S ) ≤ α ( u , x ) for some x ∈ S . α ( u , ⊤ ) = 1 and α ( u , ⊥ ) = 0 for all u ∈ U . Purbita Jana BLAST 2013 36 / 67

  51. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References L c � n -algebra L c An � n -algebra is an MV n algebra enriched by n constants. That is, it is an MV n algebra A = ( A , ∧ , ∨ , ∗ , ⊕ , → , ⊥ , 0 , 1) in which the algebra ¯ n is embedded. Purbita Jana BLAST 2013 37 / 67

  52. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References L c � n -algebra L c An � n -algebra is an MV n algebra enriched by n constants. That is, it is an MV n algebra A = ( A , ∧ , ∨ , ∗ , ⊕ , → , ⊥ , 0 , 1) in which the algebra ¯ n is embedded. L c L c � n -homomorphism is a function between two � n -algebras which preserves the operations. Purbita Jana BLAST 2013 37 / 67

  53. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBSy n n -fuzzy Boolean System ¯ An ¯ n -fuzzy Boolean System is a triple L c ( X , | = , A ) where X is a set, A is an � n -algebra and | = is an ¯ n valued fuzzy relation from X to A such that Purbita Jana BLAST 2013 38 / 67

  54. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBSy n ¯ n -fuzzy Boolean System An ¯ n -fuzzy Boolean System is a triple L c ( X , | = , A ) where X is a set, A is an � n -algebra and | = is an ¯ n valued fuzzy relation from X to A such that 1 gr ( x | = a ∗ b ) = max (0 , gr ( x | = a ) + gr ( x | = b ) − 1) Purbita Jana BLAST 2013 38 / 67

  55. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBSy n ¯ n -fuzzy Boolean System An ¯ n -fuzzy Boolean System is a triple L c ( X , | = , A ) where X is a set, A is an � n -algebra and | = is an ¯ n valued fuzzy relation from X to A such that 1 gr ( x | = a ∗ b ) = max (0 , gr ( x | = a ) + gr ( x | = b ) − 1) 2 gr ( x | = a ⊥ ) = 1 − gr ( x | = a ) Purbita Jana BLAST 2013 38 / 67

  56. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBSy n ¯ n -fuzzy Boolean System An ¯ n -fuzzy Boolean System is a triple L c ( X , | = , A ) where X is a set, A is an � n -algebra and | = is an ¯ n valued fuzzy relation from X to A such that 1 gr ( x | = a ∗ b ) = max (0 , gr ( x | = a ) + gr ( x | = b ) − 1) 2 gr ( x | = a ⊥ ) = 1 − gr ( x | = a ) 3 gr ( x | = r ) = r for all r ∈ ¯ n Purbita Jana BLAST 2013 38 / 67

  57. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBSy n ¯ n -fuzzy Boolean System An ¯ n -fuzzy Boolean System is a triple L c ( X , | = , A ) where X is a set, A is an � n -algebra and | = is an ¯ n valued fuzzy relation from X to A such that 1 gr ( x | = a ∗ b ) = max (0 , gr ( x | = a ) + gr ( x | = b ) − 1) 2 gr ( x | = a ⊥ ) = 1 − gr ( x | = a ) 3 gr ( x | = r ) = r for all r ∈ ¯ n 4 x 1 � = x 2 ⇒ gr ( x 1 | = a ) � = gr ( x 2 | = a ) for some a ∈ A Purbita Jana BLAST 2013 38 / 67

  58. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Continuous map = ′ , B ) be ¯ Let D = ( X , | = , A ) and E = ( Y , | n -fuzzy Boolean Systems. Purbita Jana BLAST 2013 39 / 67

  59. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References n -fuzzy Boolean System ¯ Continuous map = ′ , B ) be ¯ Let D = ( X , | = , A ) and E = ( Y , | n -fuzzy Boolean Systems.A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, Purbita Jana BLAST 2013 39 / 67

  60. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Continuous map = ′ , B ) be ¯ Let D = ( X , | = , A ) and E = ( Y , | n -fuzzy Boolean Systems.A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, 1 f 1 : X − → Y is a function. Purbita Jana BLAST 2013 39 / 67

  61. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Continuous map = ′ , B ) be ¯ Let D = ( X , | = , A ) and E = ( Y , | n -fuzzy Boolean Systems.A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, 1 f 1 : X − → Y is a function. 2 f 2 : B − L c → A is � n -homomorphism and Purbita Jana BLAST 2013 39 / 67

  62. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Continuous map = ′ , B ) be ¯ Let D = ( X , | = , A ) and E = ( Y , | n -fuzzy Boolean Systems.A continuous map f : D − → E is a pair ( f 1 , f 2 ) where, 1 f 1 : X − → Y is a function. 2 f 2 : B − L c → A is � n -homomorphism and = ′ b ), for all x ∈ X , b ∈ B . 3 gr ( x | = f 2 ( b )) = gr ( f 1 ( x ) | Purbita Jana BLAST 2013 39 / 67

  63. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Identity map Let D = ( X , | = , A ) be ¯ n -fuzzy Boolean System. The identity map I D : D − → D is the pair ( I 1 , I 2 ) of identity maps- Purbita Jana BLAST 2013 40 / 67

  64. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Identity map Let D = ( X , | = , A ) be ¯ n -fuzzy Boolean System. The identity map I D : D − → D is the pair ( I 1 , I 2 ) of identity maps- I 1 : X − → X I 2 : A − → A Purbita Jana BLAST 2013 40 / 67

  65. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References n -fuzzy Boolean System ¯ Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. Purbita Jana BLAST 2013 41 / 67

  66. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by Purbita Jana BLAST 2013 41 / 67

  67. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by g 1 ◦ f 1 : X − → Z f 2 ◦ g 2 : C − → A Purbita Jana BLAST 2013 41 / 67

  68. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System Composition = ′ , A ), E = ( Y , | = ′′ , B ), F = ( Z , | = ′′′ , C ). Let Let D = ( X , | ( f 1 , f 2 ) : D − → E and ( g 1 , g 2 ) : E − → F be continuous maps. The composition ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) : D − → F is defined by g 1 ◦ f 1 : X − → Z f 2 ◦ g 2 : C − → A i.e ( g 1 , g 2 ) ◦ ( f 1 , f 2 ) = ( g 1 ◦ f 1 , f 2 ◦ g 2 ). Purbita Jana BLAST 2013 41 / 67

  69. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References ¯ n -fuzzy Boolean System FBSy n ¯ n -fuzzy Boolean Systems together with continuous functions forms the category ¯ n -fuzzy Boolean System( FBSy n ). Purbita Jana BLAST 2013 42 / 67

  70. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBS n ¯ n -fuzzy Boolean Space For an ¯ n -fuzzy topological space ( X , τ ) is called an ¯ n -fuzzy Boolean space iff ( X , τ ) is zero dimensional, compact and Kolmogorov. Purbita Jana BLAST 2013 43 / 67

  71. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References FBS n n -fuzzy Boolean Space ¯ For an ¯ n -fuzzy topological space ( X , τ ) is called an ¯ n -fuzzy Boolean space iff ( X , τ ) is zero dimensional, compact and Kolmogorov. FBS n ¯ n -fuzzy topological space ( X , τ ) together with continuous map forms the category FBS n . Purbita Jana BLAST 2013 43 / 67

  72. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References L c � n -Alg L c � n -Alg L c L c � n -algebra together with � n -Alg homomorphisms form the category L c � n -Alg. Purbita Jana BLAST 2013 44 / 67

  73. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Ext where ext ( a ) : X − → ¯ n s.t. ext ( a )( x ) = gr ( x | = a ) and ext ( A ) = { ext ( a ) } a ∈ A . Purbita Jana BLAST 2013 45 / 67

  74. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References J Purbita Jana BLAST 2013 46 / 67

  75. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References fm Purbita Jana BLAST 2013 47 / 67

  76. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References S L c where Hom ( A , ¯ n ) = { � n hom v : A − → ¯ n } and gr ( v | = ∗ a ) = v ( a ). Purbita Jana BLAST 2013 48 / 67

  77. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . Purbita Jana BLAST 2013 49 / 67

  78. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . Purbita Jana BLAST 2013 49 / 67

  79. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results 1 Ext is the right adjoint to the functor J . 2 fm is the left adjoint to the functor S . 3 Ext ◦ S is the right adjoint to the functor fm ◦ J . Purbita Jana BLAST 2013 49 / 67

  80. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results L c � n − Alg is dually equivalent to the category FBSy n . 1 Purbita Jana BLAST 2013 50 / 67

  81. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results L c � n − Alg is dually equivalent to the category FBSy n . 1 2 FBS n is equivalent to the category FBSy n . Purbita Jana BLAST 2013 50 / 67

  82. Interrelation among Top Sys, Top and Frm Interrelation among Fuzzy Top Sys, Fuzzy Top and Frm L c Interrelation among FBSy n , FBS n and � n -Alg Categories Interrelation among F -Top Sys, F -Top and Frm Functors Future Direction References Results L c � n − Alg is dually equivalent to the category FBSy n . 1 2 FBS n is equivalent to the category FBSy n . L c � n − Alg is dually equivalent to the category FBS n .. 3 Purbita Jana BLAST 2013 50 / 67

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