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)

A

:= AAn, R(m)

A

:= P(Am), OA :=

• n∈N

O(n)

A ,

RA :=

• m∈N

R(m)

A ,

cO(n)

X

:= C(X n; X), cR(m)

X

:= {̺ ⊆ Am | ̺ closed in X m}, cOX :=

• n∈N

cO(n)

X ,

cRX :=

• m∈N

cR(m)

X

.

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)

A

:= AAn, R(m)

A

:= P(Am), OA :=

• n∈N

O(n)

A ,

RA :=

• m∈N

R(m)

A ,

cO(n)

X

:= C(X n; X), cR(m)

X

:= {̺ ⊆ Am | ̺ closed in X m}, cOX :=

• n∈N

cO(n)

X ,

cRX :=

• m∈N

cR(m)

X

. For f ∈ O(n)

A

and ̺ ∈ R(m)

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)

A

:= AAn, R(m)

A

:= P(Am), OA :=

• n∈N

O(n)

A ,

RA :=

• m∈N

R(m)

A ,

cO(n)

X

:= C(X n; X), cR(m)

X

:= {̺ ⊆ Am | ̺ closed in X m}, cOX :=

• n∈N

cO(n)

X ,

cRX :=

• m∈N

cR(m)

X

. For f ∈ O(n)

A

and ̺ ∈ R(m)

A ,

f ✄ ̺ :⇐ ⇒ ∀r0, . . . , rn−1 ∈ ̺ : f ◦ r0, . . . , rn−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 ⊆ cOX,

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

The Galois connection cPol-cInv (cont’d.)

For F ⊆ cOX, cInvA, T, F := cInvX F := {̺ ∈ cRX | ∀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 ⊆ cOX, cInvA, T, F := cInvX F := {̺ ∈ cRX | ∀f ∈ F : f ✄ ̺}, for Q ⊆ cRX,

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

The Galois connection cPol-cInv (cont’d.)

For F ⊆ cOX, cInvA, T, F := cInvX F := {̺ ∈ cRX | ∀f ∈ F : f ✄ ̺}, for Q ⊆ cRX, cPolA, T, Q := cPolX Q := {f ∈ cOX | ∀̺ ∈ 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 ⊆ cOX, cInvA, T, F := cInvX F := {̺ ∈ cRX | ∀f ∈ F : f ✄ ̺}, for Q ⊆ cRX, cPolA, T, Q := cPolX Q := {f ∈ cOX | ∀̺ ∈ Q : f ✄ ̺}. How can we describe the closure system induced by this Galois connection? continuous operations

cInv

• closed relations

cPol

• 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 ⊆ OA is called clone of operations on A if (1) F contains all projections, (2) for m, n ∈ N, f ∈ F (n), f0, . . . , fn−1 ∈ F (m), we also have f ◦ f0, . . . , fn−1 ∈ F.

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

Clones of operations

Reminder

A set F ⊆ OA is called clone of operations on A if (1) F contains all projections, (2) for m, n ∈ N, f ∈ F (n), f0, . . . , fn−1 ∈ F (m), we also have f ◦ f0, . . . , fn−1 ∈ F. For any set F ⊆ OA, 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 ⊆ OA is called clone of operations on A if (1) F contains all projections, (2) for m, n ∈ N, f ∈ F (n), f0, . . . , fn−1 ∈ F (m), we also have f ◦ f0, . . . , fn−1 ∈ F. For any set F ⊆ OA, the smallest clone on A containing F is denoted by Clo(F). Obviously, cOX 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 ⊆ cRX 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

Clones of closed relations

Definition

A set Q ⊆ cRX is called clone of closed relations on X if Q is closed w.r.t. general superposition of closed relations, that is: Whenever I is a set, Y = (B, S) a topological space, m, mi ∈ N, ϕ : m → B, ϕi : mi → B and ̺i ∈ Q(mi) for i ∈ I, then ϕ,Y ,X

(ϕi)i∈I (̺i)i∈I X m

∈ Q, where ϕ,Y ,X

(ϕi)i∈I (̺i)i∈I := {r ◦ ϕ | r ∈ C(Y ; X), ∀i ∈ I : r ◦ ϕi ∈ ̺i}.

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

Clones of closed relations

Definition

A set Q ⊆ cRX is called clone of closed relations on X if Q is closed w.r.t. general superposition of closed relations, that is: Whenever I is a set, Y = (B, S) a topological space, m, mi ∈ N, ϕ : m → B, ϕi : mi → B and ̺i ∈ Q(mi) for i ∈ I, then ϕ,Y ,X

(ϕi)i∈I (̺i)i∈I X m

∈ Q, where ϕ,Y ,X

(ϕi)i∈I (̺i)i∈I := {r ◦ ϕ | r ∈ C(Y ; X), ∀i ∈ I : r ◦ ϕi ∈ ̺i}.

For any set Q ⊆ cRX, the smallest clone of closed relations on X containing Q is denoted by CLO(Q).

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

Local closure operators

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

Local closure operators

Definition

For F ⊆ cOX, Q ⊆ cRX and s ∈ N:

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

Local closure operators

Definition

For F ⊆ cOX, Q ⊆ cRX and s ∈ N: s-Loc F := {f ∈ cO(n)

X

| n ∈ N, ∀a ∈ (An)s, U ∈ T s : [f (a0) ∈ U0, . . . , f (as−1) ∈ Us−1] ⇒ [∃g ∈ F : g(a0) ∈ U0, . . . , g(as−1) ∈ Us−1]}, s-LOC Q := {̺ ∈ cOX | ∀σ ⊆ ̺, |σ| ≤ s : ∃̺′ ∈ Q : σ ⊆ ̺′ ⊆ ̺}, Loc F :=

• s∈N

s-Loc F, LOC Q :=

• s∈N

s-LOC Q.

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

Characterising the Galois closures

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then:

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then: (a) s-Loc Clo(F) = cPol cInv(s) F for s ∈ N.

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then: (a) s-Loc Clo(F) = cPol cInv(s) F for s ∈ N. (b) Loc Clo(F) = cPol cInv F.

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then: (a) s-Loc Clo(F) = cPol cInv(s) F for s ∈ N. (b) Loc Clo(F) = cPol cInv F.

Theorem

Let Q ⊆ cRX. Then:

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then: (a) s-Loc Clo(F) = cPol cInv(s) F for s ∈ N. (b) Loc Clo(F) = cPol cInv F.

Theorem

Let Q ⊆ cRX. Then: (a) s-LOC CLO(Q) = cInv cPol(s) Q for s ∈ N.

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

Characterising the Galois closures

Theorem

Let F ⊆ cOX. Then: (a) s-Loc Clo(F) = cPol cInv(s) F for s ∈ N. (b) Loc Clo(F) = cPol cInv F.

Theorem

Let Q ⊆ cRX. Then: (a) s-LOC CLO(Q) = cInv cPol(s) Q for s ∈ N. (b) LOC CLO(Q) = cInv cPol Q.

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

Topologising RST

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

Topologising RST

A localisation theory consists of three main ingredients.

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

Topologising RST

A localisation theory consists of three main ingredients. (1) Localisation: Restricting the structure to suitable subsets.

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

Topologising RST

A localisation theory consists of three main ingredients. (1) Localisation: Restricting the structure to suitable subsets. (2) Classification: Calculating locally.

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

Topologising RST

A localisation theory consists of three main ingredients. (1) Localisation: Restricting the structure to suitable subsets. (2) Classification: Calculating locally. (3) Globalisation: Combining local results into global results.

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

Topologising RST

A localisation theory consists of three main ingredients. (1) Localisation: Restricting the structure to suitable subsets. (2) Classification: Calculating locally. (3) Globalisation: Combining local results into global results. What are the suitable subsets for this kind of localisation theory?

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

Finding suitable subsets

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

Finding suitable subsets

Let A = A; T; F be a topological algebra. For U ⊆ A, EA(U) :=

• e
• e ∈ Loc Clo(1)(F), im e ⊆ U
• .

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

Finding suitable subsets

Let A = A; T; F be a topological algebra. For U ⊆ A, EA(U) :=

• e
• e ∈ Loc Clo(1)(F), im e ⊆ U
• .

Lemma

Let U ⊆ A. The following are equivalent:

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

Finding suitable subsets

Let A = A; T; F be a topological algebra. For U ⊆ A, EA(U) :=

• e
• e ∈ Loc Clo(1)(F), im e ⊆ U
• .

Lemma

Let U ⊆ A. The following are equivalent: (a) ·↾U : cInv A → cR(U,TU), ̺ → ̺↾U := ̺ ∩ Uar ̺ is a homomorphism between clones of closed relations.

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

Finding suitable subsets

Let A = A; T; F be a topological algebra. For U ⊆ A, EA(U) :=

• e
• e ∈ Loc Clo(1)(F), im e ⊆ U
• .

Lemma

Let U ⊆ A. The following are equivalent: (a) ·↾U : cInv A → cR(U,TU), ̺ → ̺↾U := ̺ ∩ Uar ̺ is a homomorphism between clones of closed relations. (b) idU ∈ Loc

• e|U

U |e ∈ EA(U)}.

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

Finding suitable subsets

Let A = A; T; F be a topological algebra. For U ⊆ A, EA(U) :=

• e
• e ∈ Loc Clo(1)(F), im e ⊆ U
• .

Lemma

Let U ⊆ A. The following are equivalent: (a) ·↾U : cInv A → cR(U,TU), ̺ → ̺↾U := ̺ ∩ Uar ̺ is a homomorphism between clones of closed relations. (b) idU ∈ Loc

• e|U

U |e ∈ EA(U)}.

Additionally, if (a) holds, then [Q]↾U := {̺↾U | ̺ ∈ Q} is locally closed for every locally closed clone of closed relations Q ⊆ cInv A.

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

Restricting algebras to neighbourhoods

A

✿✿

A = A, T, cInv A cInv A, T, Loc Clo (A) cPol ≡top cInv

✿✿

A↾U := U, TU, [cInv A]↾U

↾U

U, TU, cPol [cInv A]↾U cPol cInv Definition (neighbourhood) U ∈ Neigh A :⇐ ⇒ idU ∈ Loc

• e|U

U

• e ∈ Loc Clo(1) (A) , im e ⊆ U
• |U

:=

A|U

=

• U, TU, Loc
• f |U

Uar f

• f ∈ Loc Clo (A) , im f ⊆ U
• =
• U, TU, Loc
• f |U

Uar f

• f ∈ Loc Clo (A) , f ⊲ U
• Friedrich Martin Schneider

Technische Universit¨ at Dresden A relational localisation theory for topological algebras

How many neighbourhoods are “enough”?

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

How many neighbourhoods are “enough”?

Definition

Let U ⊆ Neigh A. (1) U is called cover of A if [∀U ∈ U : ̺↾U = σ↾U] ⇒ ̺ = σ for all ̺, σ ∈ cInv A.

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

How many neighbourhoods are “enough”?

Definition

Let U ⊆ Neigh A. (1) U is called cover of A if [∀U ∈ U : ̺↾U = σ↾U] ⇒ ̺ = σ for all ̺, σ ∈ cInv A. (2) U is called c-cover of A if it is a cover of A and every U ∈ U is closed w.r.t. T.

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

How many neighbourhoods are “enough”?

Definition

Let U ⊆ Neigh A. (1) U is called cover of A if [∀U ∈ U : ̺↾U = σ↾U] ⇒ ̺ = σ for all ̺, σ ∈ cInv A. (2) U is called c-cover of A if it is a cover of A and every U ∈ U is closed w.r.t. T. Moreover, let EA(U) :=

• {EA(U) | U ∈ U}.

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

Globalisation

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

Globalisation

Theorem

Let U ⊆ Neigh A. The following are equivalent:

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

Globalisation

Theorem

Let U ⊆ Neigh A. The following are equivalent: (a) U is a cover of A.

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

Globalisation

Theorem

Let U ⊆ Neigh A. The following are equivalent: (a) U is a cover of A. (b) idA ∈ Loc EA(U)AA.

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

Globalisation

Theorem

Let U ⊆ Neigh A. The following are equivalent: (a) U is a cover of A. (b) idA ∈ Loc EA(U)AA. (c) There is an index set Φ and a map B

✿ : Φ → {A ✿↾U | U ∈ U}

such that A

✿ is approximately a retract of

ϕ∈Φ B

✿(ϕ), i.e.

there exists M : A

✿ →

ϕ∈Φ B

✿(ϕ) with

idA ∈ Loc

• Λ ◦ M
• Λ :

ϕ∈Φ B

✿(ϕ) → A ✿

• .

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

What about the example?

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

What about the example?

Reminder

Let R = R, +, −, ·, 0 be a ring. (1) e, f ∈ Id R orthogonal :⇐ ⇒ e · f = f · e = 0.

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

What about the example?

Reminder

Let R = R, +, −, ·, 0 be a ring. (1) e, f ∈ Id R orthogonal :⇐ ⇒ e · f = f · e = 0. (2) E ⊆ Id R orthogonal :⇐ ⇒ any two distinct elements of E are orthogonal.

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

What about the example?

Reminder

Let R = R, +, −, ·, 0 be a ring. (1) e, f ∈ Id R orthogonal :⇐ ⇒ e · f = f · e = 0. (2) E ⊆ Id R orthogonal :⇐ ⇒ any two distinct elements of E are orthogonal. (3) e ∈ Id R primitive :⇐ ⇒ e = 0 and for any two orthogonal idempotents f1, f2 ∈ Id R such that e = f1 + f2 it follows f1 = 0 or f2 = 0.

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

What about the example?

Reminder

Let R = R, +, −, ·, 0 be a ring. (1) e, f ∈ Id R orthogonal :⇐ ⇒ e · f = f · e = 0. (2) E ⊆ Id R orthogonal :⇐ ⇒ any two distinct elements of E are orthogonal. (3) e ∈ Id R primitive :⇐ ⇒ e = 0 and for any two orthogonal idempotents f1, f2 ∈ Id R such that e = f1 + f2 it follows f1 = 0 or f2 = 0.

Theorem (Gabriel, 1962)

Let R = R, S, +, −, ·, 0, 1 be a compact Hausdorff topological ring, 0 = 1. Then there exists an orthogonal set E ⊆ Id R of primitive idempotents such that 1 =

e∈E e.

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

Theorem

Let U ∈ cNeigh M, |U| > 1, and m ∈ N, ̺, σ ∈ cInv(m) M such that ̺↾U = σ↾U. TFAE:

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

Theorem

Let U ∈ cNeigh M, |U| > 1, and m ∈ N, ̺, σ ∈ cInv(m) M such that ̺↾U = σ↾U. TFAE: (i) Every c-cover of M|U contains U.

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

Theorem

Let U ∈ cNeigh M, |U| > 1, and m ∈ N, ̺, σ ∈ cInv(m) M such that ̺↾U = σ↾U. TFAE: (i) Every c-cover of M|U contains U. (ii) U ∈ Min⊆((cNeigh M) \ {{0}}).

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

Theorem

Let U ∈ cNeigh M, |U| > 1, and m ∈ N, ̺, σ ∈ cInv(m) M such that ̺↾U = σ↾U. TFAE: (i) Every c-cover of M|U contains U. (ii) U ∈ Min⊆((cNeigh M) \ {{0}}). (iii) U ∈ Min⊆{V ∈ cNeigh M | ̺↾V = σ↾V }.

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

Seriously, what about the example?

R = R, S, +, −, ·, 0, 1 compact Hausdorff topological ring, M = M, T, +, −, 0, (λ(r))r∈R Hausdorff topological R-module.

Lemma

cNeigh M = {im λ(e) | e ∈ Id R}.

Theorem

Let U ∈ cNeigh M, |U| > 1, and m ∈ N, ̺, σ ∈ cInv(m) M such that ̺↾U = σ↾U. TFAE: (i) Every c-cover of M|U contains U. (ii) U ∈ Min⊆((cNeigh M) \ {{0}}). (iii) U ∈ Min⊆{V ∈ cNeigh M | ̺↾V = σ↾V }. (iv) There exists a primitive idempotent e ∈ Id R such that U = im λ(e).

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

The very last slide

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

The very last slide

Thank you for your attention!!

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

References:

1 Keith A. Kearnes. Tame Congruence Theory is a localization theory. Lecture Notes from “A Course in Tame Congruence Theory” Workshop, Budapest, 2001. 2 Keith A. Kearnes, LeAnne Conaway. Minimal sets in finite rings. Algebra Universalis 51 (2004), 81–109. 3 Mike Behrisch. Relational Tame Congruence Theory and Subalgebra Primal Algebras. Master’s thesis, Dresden University of Technology, 2009. 4 Mihail Ursul. Topological Rings Satisfying Compactness Conditions. Springer 2002.

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