Sublocales of d-frames Anna Laura Suarez University of Birmingham - - PowerPoint PPT Presentation

Sublocales of d-frames

Anna Laura Suarez

University of Birmingham axs1431@cs.bham.ac.uk

September 28, 2018

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 1 / 18

Overview

1

Bitopological spaces Intuition and motivation The category BiTop

2

D-frames Intuition and motivation The category dFrm

3

Sublocales of d-frames The general case Concrete examples

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 2 / 18

Bitopological spaces

Some topologies naturally arise as join of two other ones. The Euclidian topology on R is generated by the lower open intervals {(−∞, r) : r ∈ R} and the upper ones {(r, ∞) : r ∈ R}.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 3 / 18

Bitopological spaces

Some topologies naturally arise as join of two other ones. The Euclidian topology on R is generated by the lower open intervals {(−∞, r) : r ∈ R} and the upper ones {(r, ∞) : r ∈ R}. For a Priestley space (X, ≤) the topology is the join of two spectral spaces: the ones of open upsets and open downsets.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 3 / 18

Bitopological spaces

Some topologies naturally arise as join of two other ones. The Euclidian topology on R is generated by the lower open intervals {(−∞, r) : r ∈ R} and the upper ones {(r, ∞) : r ∈ R}. For a Priestley space (X, ≤) the topology is the join of two spectral spaces: the ones of open upsets and open downsets. The Vietoris hyperspace VX of a compact Hausdorff space X has as underlying set the closed subsets {X\U : U ∈ Ω(X)}.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 3 / 18

Bitopological spaces

Some topologies naturally arise as join of two other ones. The Euclidian topology on R is generated by the lower open intervals {(−∞, r) : r ∈ R} and the upper ones {(r, ∞) : r ∈ R}. For a Priestley space (X, ≤) the topology is the join of two spectral spaces: the ones of open upsets and open downsets. The Vietoris hyperspace VX of a compact Hausdorff space X has as underlying set the closed subsets {X\U : U ∈ Ω(X)}. The topology is the join of upper and lower topologies, with bases:

U = {C ∈ VX : C ⊆ U}. ♦U = {C ∈ VX : C ∩ U = ∅}

Where U varies over Ω(X).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 3 / 18

Bitopological spaces

A bitopological space is a structure (X, τ +, τ −) where X is a set and τ + and τ − two topologies on it. We call τ + the upper, or positive, topology. We call τ − the negative, or lower, topology. The category BiTop has bitopological spaces as objects, bicontinuous functions as maps.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 4 / 18

D-frames: intuition

D-frames are quadruples (L+, L−, con, tot) where L+ and L− are frames, and con, tot ⊆ L+ × L−; satisfying some axioms. The intuition is: L+ and L− are the frames of positive and negative opens respectively. The pairs of opens in con are the disjoint pairs. The pairs of opens in tot are the covering pairs (i.e. those whose union covers the whole space).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 5 / 18

D-frames: example

For any two frames L+ and L− we can set con and tot to be as small as the axiom allow. That is we set:

x+x− ∈ con if and only if x+ = 0+ or x− = 0−. x+x− ∈ tot if and only if x+ = 1+ or x− = 1−.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 6 / 18

D-frames: example

For any two frames L+ and L− we can set con and tot to be as small as the axiom allow. That is we set:

x+x− ∈ con if and only if x+ = 0+ or x− = 0−. x+x− ∈ tot if and only if x+ = 1+ or x− = 1−.

The following is a bitopological space with its d-frame of opens.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 6 / 18

D-frames: the two orders

On the product L+ × L− we have: The information order ⊑: we define a+a− ⊑ b+b− if and only if a+ ≤ b+ and a− ≤ b−. The logical order ≤: we define a+a− ≤ b+b− if and only if a+ ≤ b+ and b− ≤ a−.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 7 / 18

D-frames: axioms

A quadruple (L+, L−, con, tot) where L+ and L− are frames and con, tot ⊆ L+ × L− is a d-frame if the following four axioms hold: (D1) con is a ⊑-downset and tot is a ⊑-upset.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 8 / 18

D-frames: axioms

A quadruple (L+, L−, con, tot) where L+ and L− are frames and con, tot ⊆ L+ × L− is a d-frame if the following four axioms hold: (D1) con is a ⊑-downset and tot is a ⊑-upset. (D2) con and tot are ≤-sublattices. In particular 1+0−, 0+1− ∈ con ∩ tot. (D3) The set con is Scott closed.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 8 / 18

D-frames: axioms

A quadruple (L+, L−, con, tot) where L+ and L− are frames and con, tot ⊆ L+ × L− is a d-frame if the following four axioms hold: (D1) con is a ⊑-downset and tot is a ⊑-upset. (D2) con and tot are ≤-sublattices. In particular 1+0−, 0+1− ∈ con ∩ tot. (D3) The set con is Scott closed. (D4). Whenever a+b− ∈ con and a+c− ∈ tot we have b− ≤ c−. Similarly whenever b+a− ∈ con and c+a− ∈ tot we have b+ ≤ c+.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 8 / 18

The category dFrm

The category dFrm has d-frames as objects. A morphism f : (L+, L−, conL, totL) → (M+, M−, conM, totM) is defined to be a pair of frame maps (f +, f −) : (L+, L−) → (M+, M−) such that f + × f − : L+ × L− → M+ × M− preserves con and tot.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 9 / 18

Pseudocomplements

Let L = (L+, L−con, tot) be a d-frame.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 10 / 18

Pseudocomplements

Let L = (L+, L−con, tot) be a d-frame.

Definition

For a+ ∈ L+, the element ∼a+ := {x− ∈ L− : a+x− ∈ con} is the pseudocomplement of a+.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 10 / 18

Pseudocomplements

Let L = (L+, L−con, tot) be a d-frame.

Definition

For a+ ∈ L+, the element ∼a+ := {x− ∈ L− : a+x− ∈ con} is the pseudocomplement of a+. This is a complement if (a+, ∼a+) ∈ tot. Pseudocomplementation of elements of L− is defined similarly.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 10 / 18

Pseudocomplements

Let L = (L+, L−con, tot) be a d-frame.

Definition

For a+ ∈ L+, the element ∼a+ := {x− ∈ L− : a+x− ∈ con} is the pseudocomplement of a+. This is a complement if (a+, ∼a+) ∈ tot. Pseudocomplementation of elements of L− is defined similarly.

Definition

L is Boolean if every element from L+ and L− is complemented.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 10 / 18

Recall: monotopological sublocales

For a frame L the following are interdefinable: Extremal epimorphisms (in Frm) from L. Frame surjections from L. Congruences on L.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 11 / 18

Recall: monotopological sublocales

For a frame L the following are interdefinable: Extremal epimorphisms (in Frm) from L. Frame surjections from L. Congruences on L. Given any relation R on L we can compute the smallest congruence containing it. This gives a quotient qR : L ։ L/R.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 11 / 18

Bitopological sublocales

Let L = (L+, L−, con, tot) be a d-frame.

Definition

Let (C +, C −) be a pair of congruences where C ± is on L±. Consider the quotient map qC : L+ × L− ։ (L+/C +) × (L−/C −).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 12 / 18

Bitopological sublocales

Let L = (L+, L−, con, tot) be a d-frame.

Definition

Let (C +, C −) be a pair of congruences where C ± is on L±. Consider the quotient map qC : L+ × L− ։ (L+/C +) × (L−/C −). The pair (C +, C −) is called reasonable if the structure (L+/C +, L−/C −, qC[con], qC[tot]) is a d-frame.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 12 / 18

Bitopological sublocales

Let L = (L+, L−, con, tot) be a d-frame.

Definition

Let (C +, C −) be a pair of congruences where C ± is on L±. Consider the quotient map qC : L+ × L− ։ (L+/C +) × (L−/C −). The pair (C +, C −) is called reasonable if the structure (L+/C +, L−/C −, qC[con], qC[tot]) is a d-frame. We have a theorem. The following are interdefinable: Extremal epimorphisms (in dFrm) from L. D-frame surjections s : L ։ M satisfying some extra conditions. Reasonable pairs of congruences (C +, C −) on (L+, L−). .

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 12 / 18

Bitopological sublocales

Let L = (L+, L−, con, tot) be a d-frame.

Definition

Let (C +, C −) be a pair of congruences where C ± is on L±. Consider the quotient map qC : L+ × L− ։ (L+/C +) × (L−/C −). The pair (C +, C −) is called reasonable if the structure (L+/C +, L−/C −, qC[con], qC[tot]) is a d-frame. We have a theorem. The following are interdefinable: Extremal epimorphisms (in dFrm) from L. D-frame surjections s : L ։ M satisfying some extra conditions. Reasonable pairs of congruences (C +, C −) on (L+, L−). . Given a pair of relations (R+, R−) where R± is on L±, we can compute the smallest reasonable congruence pair containing it. This gives a quotient qR : L ։ L/R in dFrm. However, this is difficult to compute.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 12 / 18

Bitopological sublocales

Changing the starting relations (R+, R−) gives different kinds of

• sublocales. For a+ ∈ L+ we want to know what are the reasonable

congruence pairs that the following induce. (R(op(a+)), id−) (positive open sublocale).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 13 / 18

Bitopological sublocales

Changing the starting relations (R+, R−) gives different kinds of

• sublocales. For a+ ∈ L+ we want to know what are the reasonable

congruence pairs that the following induce. (R(op(a+)), id−) (positive open sublocale). (R(cl(a+)), id−) (positive closed sublocale).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 13 / 18

Bitopological sublocales

Changing the starting relations (R+, R−) gives different kinds of

• sublocales. For a+ ∈ L+ we want to know what are the reasonable

congruence pairs that the following induce. (R(op(a+)), id−) (positive open sublocale). (R(cl(a+)), id−) (positive closed sublocale). (R∼∼, R∼∼) (double pseudocomplementation). Here R∼∼ identifies a+ and b+ precisely when ∼∼a+ = ∼∼b+, similarly for elements of L−.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 13 / 18

Results: open and closed sublocales

Given a d-frame (L+, L−, con, tot) and some a+a− ∈ L+ × L− we have the following.

Proposition

Whenever L+ is linear, or L Boolean, or con and tot are minimal, (R+(op(a+)), id−) induces (R+(op(a+)), R−(cl(∼a+))).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 14 / 18

Results: open and closed sublocales

Given a d-frame (L+, L−, con, tot) and some a+a− ∈ L+ × L− we have the following.

Proposition

Whenever L+ is linear, or L Boolean, or con and tot are minimal, (R+(op(a+)), id−) induces (R+(op(a+)), R−(cl(∼a+))). That is, every positive open sublocale induces the negative closed sublocale of its pseudocomplement.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 14 / 18

Results: open and closed sublocales

Given a d-frame (L+, L−, con, tot) and some a+a− ∈ L+ × L− we have the following.

Proposition

Whenever L+ is linear, or L Boolean, or con and tot are minimal, (R+(op(a+)), id−) induces (R+(op(a+)), R−(cl(∼a+))). That is, every positive open sublocale induces the negative closed sublocale of its pseudocomplement.

Proposition

Whenever a+a− is a complemented pair,the relations (R+(op(a+)), id− and (R−(cl(a−)), id− both induce the reasonable pair of congruences (R+(op(a+)), R−(cl(a−))).

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 14 / 18

Results: open and closed sublocales

Given a d-frame (L+, L−, con, tot) and some a+a− ∈ L+ × L− we have the following.

Proposition

Whenever L+ is linear, or L Boolean, or con and tot are minimal, (R+(op(a+)), id−) induces (R+(op(a+)), R−(cl(∼a+))). That is, every positive open sublocale induces the negative closed sublocale of its pseudocomplement.

Proposition

Whenever a+a− is a complemented pair,the relations (R+(op(a+)), id− and (R−(cl(a−)), id− both induce the reasonable pair of congruences (R+(op(a+)), R−(cl(a−))). That is, every open sublocale induces the negative closed sublocale of its complement and vice-versa.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 14 / 18

Results: double pseudocomplementation

Consider the map ∼∼ : L+ → L+ as a+ → ∼∼a+. Similarly for L−. This always is a closure operator.

Proposition

Whenever ∼∼ preserves finite meets, the relation B induces itself. This happens whenever L is Boolean or linear.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 15 / 18

A partial Booleanization

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 16 / 18

A partial Booleanization

Define totM :=↑ ({((a+, ∼a+) : a+ ∈ L+} ∪ {∼a−, a−) : a− ∈ L−}). Let HdFrm be the subcategory of dFrm of d-frames and pseudocomplement-preserving maps.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 16 / 18

A partial Booleanization

Define totM :=↑ ({((a+, ∼a+) : a+ ∈ L+} ∪ {∼a−, a−) : a− ∈ L−}). Let HdFrm be the subcategory of dFrm of d-frames and pseudocomplement-preserving maps.

Proposition

Whenever ∼∼ preserves finite meets, the quotient qB : L ։ (L+/B+, L−/B−, qB[con], qB[totM]) is the Booleanization of L.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 16 / 18

A partial Booleanization

Define totM :=↑ ({((a+, ∼a+) : a+ ∈ L+} ∪ {∼a−, a−) : a− ∈ L−}). Let HdFrm be the subcategory of dFrm of d-frames and pseudocomplement-preserving maps.

Proposition

Whenever ∼∼ preserves finite meets, the quotient qB : L ։ (L+/B+, L−/B−, qB[con], qB[totM]) is the Booleanization of L. That is, any morphism f : L → C of HdFrm to a Boolean d-frame C factors through it uniquely.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 16 / 18

References

• A. Jung, M. A. Moshier (2006)

On the bitopological nature of Stone duality Preprint.

• T. Jakl (2018)

D-frames as algebraic duals of bitopological spaces PhD thesis, University of Birmingham.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 17 / 18

Summary

D-frames are order-theoretical duals of bitopological spaces.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 18 / 18

Summary

D-frames are order-theoretical duals of bitopological spaces. Computing the sublocale (extremal epi) induced by a pair of relations takes transfinitely many steps in general.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 18 / 18

Summary

D-frames are order-theoretical duals of bitopological spaces. Computing the sublocale (extremal epi) induced by a pair of relations takes transfinitely many steps in general. However in several cases open, closed, and double pseudocomplementation sublocales are easy to compute.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 18 / 18

Summary

D-frames are order-theoretical duals of bitopological spaces. Computing the sublocale (extremal epi) induced by a pair of relations takes transfinitely many steps in general. However in several cases open, closed, and double pseudocomplementation sublocales are easy to compute. In particular, the last one gives a bitopological Booleanization.

Anna Laura Suarez (University of Birmingham) Bitopological sublocales September 28, 2018 18 / 18