A relational localisation theory for topological algebras Friedrich - PowerPoint PPT Presentation

A relational localisation theory for topological algebras Friedrich Martin Schneider Technische Universit¨ at Dresden Novi Sad, March 17, 2012 Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. ◮ introduce a relational localisation theory for topological algebras, Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. ◮ introduce a relational localisation theory for topological algebras, identify suitable subsets, Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. ◮ introduce a relational localisation theory for topological algebras, identify suitable subsets, describe the restriction process Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. ◮ introduce a relational localisation theory for topological algebras, identify suitable subsets, describe the restriction process and explain how to reconstruct an algebra from its decomposition. Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

What will this talk be about? I will ... ◮ sketch a general Galois theory for continuous operations and closed relations on a topological space and characterise the corresponding system of Galois closures. ◮ introduce a relational localisation theory for topological algebras, identify suitable subsets, describe the restriction process and explain how to reconstruct an algebra from its decomposition. ◮ explore the developed concepts for modules of compact rings . Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv Let X = ( A , T ) be a topological space, m , n ∈ N . Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv Let X = ( A , T ) be a topological space, m , n ∈ N . O ( n ) R ( m ) := A A n , := P ( A m ) , A A O ( n ) R ( m ) � � O A := A , R A := A , n ∈ N m ∈ N := { ̺ ⊆ A m | ̺ closed in X m } , cO ( n ) cR ( m ) := C ( X n ; X ) , X X cO ( n ) cR ( m ) � � cO X := cR X := X , . X n ∈ N m ∈ N Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv Let X = ( A , T ) be a topological space, m , n ∈ N . O ( n ) R ( m ) := A A n , := P ( A m ) , A A O ( n ) R ( m ) � � O A := A , R A := A , n ∈ N m ∈ N := { ̺ ⊆ A m | ̺ closed in X m } , cO ( n ) cR ( m ) := C ( X n ; X ) , X X cO ( n ) cR ( m ) � � cO X := cR X := X , . X n ∈ N m ∈ N For f ∈ O ( n ) and ̺ ∈ R ( m ) A , A Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv Let X = ( A , T ) be a topological space, m , n ∈ N . O ( n ) R ( m ) := A A n , := P ( A m ) , A A O ( n ) R ( m ) � � O A := A , R A := A , n ∈ N m ∈ N := { ̺ ⊆ A m | ̺ closed in X m } , cO ( n ) cR ( m ) := C ( X n ; X ) , X X cO ( n ) cR ( m ) � � cO X := cR X := X , . X n ∈ N m ∈ N For f ∈ O ( n ) and ̺ ∈ R ( m ) A , A f ✄ ̺ : ⇐ ⇒ ∀ r 0 , . . . , r n − 1 ∈ ̺ : f ◦ � r 0 , . . . , r n − 1 � ∈ ̺ ̺ ∈ Sub( � A ; f � m ) ⇐ ⇒ f ∈ Hom( � A ; ̺ � n ; � A ; ̺ � ) . ⇐ ⇒ Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv (cont’d.) Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv (cont’d.) For F ⊆ cO X , Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv (cont’d.) For F ⊆ cO X , cInv � A , T , F � := cInv X F := { ̺ ∈ cR X | ∀ f ∈ F : f ✄ ̺ } , Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv (cont’d.) For F ⊆ cO X , cInv � A , T , F � := cInv X F := { ̺ ∈ cR X | ∀ f ∈ F : f ✄ ̺ } , for Q ⊆ cR X , Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

The Galois connection cPol-cInv (cont’d.) For F ⊆ cO X , cInv � A , T , F � := cInv X F := { ̺ ∈ cR X | ∀ f ∈ F : f ✄ ̺ } , for Q ⊆ cR X , cPol � A , T , Q � := cPol X Q := { f ∈ cO X | ∀ ̺ ∈ Q : f ✄ ̺ } . Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

� � The Galois connection cPol-cInv (cont’d.) For F ⊆ cO X , cInv � A , T , F � := cInv X F := { ̺ ∈ cR X | ∀ f ∈ F : f ✄ ̺ } , for Q ⊆ cR X , cPol � A , T , Q � := cPol X Q := { f ∈ cO X | ∀ ̺ ∈ Q : f ✄ ̺ } . How can we describe the closure system induced by this Galois connection? cPol continuous operations closed relations cInv Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of operations Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of operations Reminder A set F ⊆ O A is called clone of operations on A if (1) F contains all projections, (2) for m , n ∈ N , f ∈ F ( n ) , f 0 , . . . , f n − 1 ∈ F ( m ) , we also have f ◦ � f 0 , . . . , f n − 1 � ∈ F . Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of operations Reminder A set F ⊆ O A is called clone of operations on A if (1) F contains all projections, (2) for m , n ∈ N , f ∈ F ( n ) , f 0 , . . . , f n − 1 ∈ F ( m ) , we also have f ◦ � f 0 , . . . , f n − 1 � ∈ F . For any set F ⊆ O A , the smallest clone on A containing F is denoted by Clo( F ). Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of operations Reminder A set F ⊆ O A is called clone of operations on A if (1) F contains all projections, (2) for m , n ∈ N , f ∈ F ( n ) , f 0 , . . . , f n − 1 ∈ F ( m ) , we also have f ◦ � f 0 , . . . , f n − 1 � ∈ F . For any set F ⊆ O A , the smallest clone on A containing F is denoted by Clo( F ). Obviously, cO X is a clone of operations on A . Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of closed relations Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras

Clones of closed relations Definition A set Q ⊆ cR X is called clone of closed relations on X if Q is closed w.r.t. general superposition of closed relations Friedrich Martin Schneider Technische Universit¨ at Dresden A relational localisation theory for topological algebras