localic krull dimensionrmi.tsu.ge/tolo5/abstracts/lucero_bryan.pdf · point free approach 1 locale...
TRANSCRIPT
![Page 1: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/1.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull Dimension
Joel Lucero-Bryan, Khalifa University
Joint work:Guram Bezhanishvili, New Mexico State University
Nick Bezhanishvili, University of AmsterdamJan van Mill, University of Amsterdam
ToLo V, Tbilisi, Georgia13–17 June 2016
![Page 2: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/2.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation I
Definition
R: commutative ringSpec(R): set of prime ideals of RKrull dimension of R: supremum of lengths of chains in Spec(R)ordered by ⊆
Extends to:
Spectral spaces via specialization order
(Bounded) distributive lattices via Stone duality
![Page 3: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/3.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation I
Definition
R: commutative ringSpec(R): set of prime ideals of RKrull dimension of R: supremum of lengths of chains in Spec(R)ordered by ⊆
Extends to:
Spectral spaces via specialization order
(Bounded) distributive lattices via Stone duality
![Page 4: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/4.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation I
Definition
R: commutative ringSpec(R): set of prime ideals of RKrull dimension of R: supremum of lengths of chains in Spec(R)ordered by ⊆
Extends to:
Spectral spaces via specialization order
(Bounded) distributive lattices via Stone duality
![Page 5: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/5.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation I
Definition
R: commutative ringSpec(R): set of prime ideals of RKrull dimension of R: supremum of lengths of chains in Spec(R)ordered by ⊆
Extends to:
Spectral spaces via specialization order
(Bounded) distributive lattices via Stone duality
![Page 6: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/6.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 7: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/7.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 8: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/8.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 9: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/9.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 10: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/10.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 11: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/11.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 12: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/12.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Motivation II
Isbell 1985
• [Krull dimension is] “spectacularly wrong for the most popularspaces, vanishing for all non-empty Hausdorff spaces; but itseems to be the only dimension of interest for the Zariskispaces of algebraic geometry.”
Remedy: graduated dimension
Goal:
Modify Krull dimension motivated by applications in modal logic
Point free approach
1 Locale of open subsets ⇒ Heyting algebras and intuitionisticlogic
2 Power set closure algebra ⇒ modal logics above S4
![Page 13: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/13.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Closure algebras
Definition
Closure Algebra A = (A,C): Boolean algebra A with a closureoperator C : A→ A satisfying the Kuratowski axioms
C(a ∨ b) = Ca ∨ Cb CCa ≤ Ca
C0 = 0 a ≤ Ca
Interior operator: I : A→ A is dual to C; i.e. Ia = −C(−a)
Natural examples:
(℘X ,C) where ℘X is the power set of X , a topological spacewith closure operator C
(℘W ,R−1) where (W ,R) is a quasi-ordered set andR−1(A) := w ∈W | ∃v ∈ A, wRv
![Page 14: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/14.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Closure algebras
Definition
Closure Algebra A = (A,C): Boolean algebra A with a closureoperator C : A→ A satisfying the Kuratowski axioms
C(a ∨ b) = Ca ∨ Cb CCa ≤ Ca
C0 = 0 a ≤ Ca
Interior operator: I : A→ A is dual to C; i.e. Ia = −C(−a)
Natural examples:
(℘X ,C) where ℘X is the power set of X , a topological spacewith closure operator C
(℘W ,R−1) where (W ,R) is a quasi-ordered set andR−1(A) := w ∈W | ∃v ∈ A, wRv
![Page 15: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/15.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Closure algebras
Definition
Closure Algebra A = (A,C): Boolean algebra A with a closureoperator C : A→ A satisfying the Kuratowski axioms
C(a ∨ b) = Ca ∨ Cb CCa ≤ Ca
C0 = 0 a ≤ Ca
Interior operator: I : A→ A is dual to C; i.e. Ia = −C(−a)
Natural examples:
(℘X ,C) where ℘X is the power set of X , a topological spacewith closure operator C
(℘W ,R−1) where (W ,R) is a quasi-ordered set andR−1(A) := w ∈W | ∃v ∈ A, wRv
![Page 16: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/16.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Closure algebras
Definition
Closure Algebra A = (A,C): Boolean algebra A with a closureoperator C : A→ A satisfying the Kuratowski axioms
C(a ∨ b) = Ca ∨ Cb CCa ≤ Ca
C0 = 0 a ≤ Ca
Interior operator: I : A→ A is dual to C; i.e. Ia = −C(−a)
Natural examples:
(℘X ,C) where ℘X is the power set of X , a topological spacewith closure operator C
(℘W ,R−1) where (W ,R) is a quasi-ordered set andR−1(A) := w ∈W | ∃v ∈ A, wRv
![Page 17: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/17.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 18: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/18.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 19: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/19.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 20: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/20.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 21: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/21.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 22: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/22.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 23: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/23.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 24: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/24.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Heyting algebras
Definition
Heyting Algebra H: bounded distributive lattice such that ∧ hasresidual → satisfying a ≤ b → c iff a ∧ b ≤ c
Natural examples:
Open subsets of a topological space X ; a.k.a. the locale Ω(X )
Upsets of a partially ordered set
Connecting closure and Heyting algebras
Heyting algebra of open elements of A = (A,C):H(A) = Ia | a ∈ AClosure algebra associated with H: A(H) free Boolean extension ofH with ‘appropriate’ closure operatorH(A(H)) ∼= H and A(H(A)) isomorphic to subalgebra of A
![Page 25: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/25.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 26: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/26.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 27: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/27.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 28: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/28.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 29: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/29.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 30: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/30.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a closure algebra
Recall
For A = (A,C), let A∗ be the set of ultrafilters of A
Quasi-order A∗: xRy iff ∀a ∈ A, a ∈ y ⇒ Ca ∈ x
R-chain: finite sequence xi ∈ A∗ | i < n such that xiRxi+1
and xi+1Rxi for all i
length of R-chain xi | i < n is n − 1allow the empty R-chain which has length −1
Definition
The Krull dimension kdim(A) of a closure algebra A is thesupremum of the lengths of R-chains in A∗. If the supremum isnot finite, then we write kdim(A) =∞.
![Page 31: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/31.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples
Let A = (A,C) be a closure algebra:
Example 1
If A is trivial then kdim(A) = −1 (since A has no ultrafilters,A∗ = ∅)
Example 2
If C = idA then kdim(A) = 0 (since the relation for A∗ is equality)
Example 3: A = 0, a, b, 1Let Ca = Cb = 1. Then kdim(A) = 0 (since A∗ is two elementcluster)
![Page 32: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/32.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples
Let A = (A,C) be a closure algebra:
Example 1
If A is trivial then kdim(A) = −1 (since A has no ultrafilters,A∗ = ∅)
Example 2
If C = idA then kdim(A) = 0 (since the relation for A∗ is equality)
Example 3: A = 0, a, b, 1Let Ca = Cb = 1. Then kdim(A) = 0 (since A∗ is two elementcluster)
![Page 33: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/33.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples
Let A = (A,C) be a closure algebra:
Example 1
If A is trivial then kdim(A) = −1 (since A has no ultrafilters,A∗ = ∅)
Example 2
If C = idA then kdim(A) = 0 (since the relation for A∗ is equality)
Example 3: A = 0, a, b, 1Let Ca = Cb = 1. Then kdim(A) = 0 (since A∗ is two elementcluster)
![Page 34: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/34.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 35: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/35.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 36: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/36.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 37: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/37.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 38: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/38.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 39: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/39.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some easy examples cont.
Example 4: A = 0, a, b, 1Let Ca = a and Cb = 1. Then kdim(A) = 1 (since A∗ is twoelement chain)Observe: ICa = Ia = −C− a = −Cb = −1 = 0
Definitions
Let A = (A,C) be a closure algebra and a ∈ A:
a is nowhere dense in A: provided ICa = 0
Relativization Aa of A to a: the interval [0, a] with operations∧,∨ as in A, the complement of b ∈ Aa is a− b, and closureof b ∈ Aa is a ∧ Cb
![Page 40: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/40.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 41: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/41.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 42: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/42.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 43: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/43.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 44: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/44.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 45: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/45.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 46: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/46.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
kdim is not defined point free–requires A∗!
Internal definition
For a closure algebra A = (A,C),kdim(A) = −1 if A is the trivial algebra,kdim(A) ≤ n if ∀d nowhere dense in A, kdim(Ad) ≤ n − 1,kdim(A) = n if kdim(A) ≤ n and kdim(A) 6≤ n − 1,kdim(A) =∞ if kdim(A) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(A) are equivalentFinite kdim is expressible by a modal formula
![Page 47: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/47.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 48: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/48.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 49: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/49.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 50: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/50.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 51: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/51.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 52: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/52.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 53: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/53.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the modal language in CA
Interpretations in A = (A,C)
letters elements of A
Classical connectives Boolean operations of A
diamond Cbox I
Formula ϕ is valid in A: ϕ evaluates to 1 for all interpretations;written A ϕ
The bd formulas
Let n ≥ 1:
bd1 := ♦p1 → p1,
bdn+1 := ♦ (pn+1 ∧ ¬bdn)→ pn+1.
![Page 54: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/54.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for CA
Theorem
Let A be a nontrivial closure algebra and n ≥ 1. TFAE:
1 kdim(A) ≤ n − 1.
2 A bdn.
3 There does not exist a sequence e0, . . . , en of nonzero closedelements of A such that e0 = 1 and ei+1 is nowhere dense inAei for each i ∈ 0, . . . , n − 1.
4 depth(A∗) ≤ n.
![Page 55: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/55.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for CA
Theorem
Let A be a nontrivial closure algebra and n ≥ 1. TFAE:
1 kdim(A) ≤ n − 1.
2 A bdn.
3 There does not exist a sequence e0, . . . , en of nonzero closedelements of A such that e0 = 1 and ei+1 is nowhere dense inAei for each i ∈ 0, . . . , n − 1.
4 depth(A∗) ≤ n.
![Page 56: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/56.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for CA
Theorem
Let A be a nontrivial closure algebra and n ≥ 1. TFAE:
1 kdim(A) ≤ n − 1.
2 A bdn.
3 There does not exist a sequence e0, . . . , en of nonzero closedelements of A such that e0 = 1 and ei+1 is nowhere dense inAei for each i ∈ 0, . . . , n − 1.
4 depth(A∗) ≤ n.
![Page 57: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/57.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for CA
Theorem
Let A be a nontrivial closure algebra and n ≥ 1. TFAE:
1 kdim(A) ≤ n − 1.
2 A bdn.
3 There does not exist a sequence e0, . . . , en of nonzero closedelements of A such that e0 = 1 and ei+1 is nowhere dense inAei for each i ∈ 0, . . . , n − 1.
4 depth(A∗) ≤ n.
![Page 58: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/58.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a Heyting algebra
Recall
For a Heyting algebra H, let H∗ be the set of prime filtersH∗ can be partially ordered by ⊆ (closely related to R for A(H)∗)
Definition
The Krull dimension kdim(H) of a Heyting algebra H is thesupremum of the lengths of chains in H∗. If the supremum is notfinite, then we write kdim(H) =∞.
Lemma
If A is a closure algebra, then kdim(A) = kdim(H(A)).
If H is a Heyting algebra, then kdim(H) = kdim(A(H)).
![Page 59: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/59.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a Heyting algebra
Recall
For a Heyting algebra H, let H∗ be the set of prime filtersH∗ can be partially ordered by ⊆ (closely related to R for A(H)∗)
Definition
The Krull dimension kdim(H) of a Heyting algebra H is thesupremum of the lengths of chains in H∗. If the supremum is notfinite, then we write kdim(H) =∞.
Lemma
If A is a closure algebra, then kdim(A) = kdim(H(A)).
If H is a Heyting algebra, then kdim(H) = kdim(A(H)).
![Page 60: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/60.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a Heyting algebra
Recall
For a Heyting algebra H, let H∗ be the set of prime filtersH∗ can be partially ordered by ⊆ (closely related to R for A(H)∗)
Definition
The Krull dimension kdim(H) of a Heyting algebra H is thesupremum of the lengths of chains in H∗. If the supremum is notfinite, then we write kdim(H) =∞.
Lemma
If A is a closure algebra, then kdim(A) = kdim(H(A)).
If H is a Heyting algebra, then kdim(H) = kdim(A(H)).
![Page 61: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/61.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a Heyting algebra
Recall
For a Heyting algebra H, let H∗ be the set of prime filtersH∗ can be partially ordered by ⊆ (closely related to R for A(H)∗)
Definition
The Krull dimension kdim(H) of a Heyting algebra H is thesupremum of the lengths of chains in H∗. If the supremum is notfinite, then we write kdim(H) =∞.
Lemma
If A is a closure algebra, then kdim(A) = kdim(H(A)).
If H is a Heyting algebra, then kdim(H) = kdim(A(H)).
![Page 62: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/62.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Krull dimension of a Heyting algebra
Recall
For a Heyting algebra H, let H∗ be the set of prime filtersH∗ can be partially ordered by ⊆ (closely related to R for A(H)∗)
Definition
The Krull dimension kdim(H) of a Heyting algebra H is thesupremum of the lengths of chains in H∗. If the supremum is notfinite, then we write kdim(H) =∞.
Lemma
If A is a closure algebra, then kdim(A) = kdim(H(A)).
If H is a Heyting algebra, then kdim(H) = kdim(A(H)).
![Page 63: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/63.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
An example
Definition
Let H be a Heyting algebra and a ∈ H:a is dense in H: provided ¬a := a→ 0 = 0
Example 4 revisited
H = 0, b, 1 open elements from previous Example 4b is dense in A ... also in H
Definition
Let H be a Heyting algebra and a ∈ H: Relativization Ha of H toa: the interval [a, 1] with operations ∧,∨,→ as in H and bottom a
![Page 64: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/64.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
An example
Definition
Let H be a Heyting algebra and a ∈ H:a is dense in H: provided ¬a := a→ 0 = 0
Example 4 revisited
H = 0, b, 1 open elements from previous Example 4b is dense in A ... also in H
Definition
Let H be a Heyting algebra and a ∈ H: Relativization Ha of H toa: the interval [a, 1] with operations ∧,∨,→ as in H and bottom a
![Page 65: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/65.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
An example
Definition
Let H be a Heyting algebra and a ∈ H:a is dense in H: provided ¬a := a→ 0 = 0
Example 4 revisited
H = 0, b, 1 open elements from previous Example 4b is dense in A ... also in H
Definition
Let H be a Heyting algebra and a ∈ H: Relativization Ha of H toa: the interval [a, 1] with operations ∧,∨,→ as in H and bottom a
![Page 66: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/66.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
An example
Definition
Let H be a Heyting algebra and a ∈ H:a is dense in H: provided ¬a := a→ 0 = 0
Example 4 revisited
H = 0, b, 1 open elements from previous Example 4b is dense in A ... also in H
Definition
Let H be a Heyting algebra and a ∈ H: Relativization Ha of H toa: the interval [a, 1] with operations ∧,∨,→ as in H and bottom a
![Page 67: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/67.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
An example
Definition
Let H be a Heyting algebra and a ∈ H:a is dense in H: provided ¬a := a→ 0 = 0
Example 4 revisited
H = 0, b, 1 open elements from previous Example 4b is dense in A ... also in H
Definition
Let H be a Heyting algebra and a ∈ H: Relativization Ha of H toa: the interval [a, 1] with operations ∧,∨,→ as in H and bottom a
![Page 68: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/68.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
Internal definition
For a Heyting algebra H,kdim(H) = −1 if H is the trivial algebra,kdim(H) ≤ n if kdim(Hb) ≤ n − 1 for every dense b ∈ H,kdim(H) = n if kdim(H) ≤ n and kdim(H) 6≤ n − 1,kdim(H) =∞ if kdim(H) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(H) are equivalentFinite kdim(H) is expressible by an intuitionistic formula
![Page 69: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/69.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
Internal definition
For a Heyting algebra H,kdim(H) = −1 if H is the trivial algebra,kdim(H) ≤ n if kdim(Hb) ≤ n − 1 for every dense b ∈ H,kdim(H) = n if kdim(H) ≤ n and kdim(H) 6≤ n − 1,kdim(H) =∞ if kdim(H) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(H) are equivalentFinite kdim(H) is expressible by an intuitionistic formula
![Page 70: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/70.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
Internal definition
For a Heyting algebra H,kdim(H) = −1 if H is the trivial algebra,kdim(H) ≤ n if kdim(Hb) ≤ n − 1 for every dense b ∈ H,kdim(H) = n if kdim(H) ≤ n and kdim(H) 6≤ n − 1,kdim(H) =∞ if kdim(H) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(H) are equivalentFinite kdim(H) is expressible by an intuitionistic formula
![Page 71: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/71.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Internal definition
Internal definition
For a Heyting algebra H,kdim(H) = −1 if H is the trivial algebra,kdim(H) ≤ n if kdim(Hb) ≤ n − 1 for every dense b ∈ H,kdim(H) = n if kdim(H) ≤ n and kdim(H) 6≤ n − 1,kdim(H) =∞ if kdim(H) 6≤ n for any n = −1, 0, 1, 2, . . . .
Observation
Both definitions for kdim(H) are equivalentFinite kdim(H) is expressible by an intuitionistic formula
![Page 72: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/72.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the intuitionistic lang. in HA
Interpretations in H
letters elements of H
conjunction meet in H
disjunction join in H
implication → in H
Formula ϕ is valid in H: ϕ evaluates to 1 for all interpretations;written H ϕ
The ibd formulas
Let n ≥ 1:
ibd1 := p1 ∨ ¬p1,ibdn+1 := pn+1 ∨ (pn+1 → ibdn).
![Page 73: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/73.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the intuitionistic lang. in HA
Interpretations in H
letters elements of H
conjunction meet in H
disjunction join in H
implication → in H
Formula ϕ is valid in H: ϕ evaluates to 1 for all interpretations;written H ϕ
The ibd formulas
Let n ≥ 1:
ibd1 := p1 ∨ ¬p1,ibdn+1 := pn+1 ∨ (pn+1 → ibdn).
![Page 74: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/74.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Interpreting the intuitionistic lang. in HA
Interpretations in H
letters elements of H
conjunction meet in H
disjunction join in H
implication → in H
Formula ϕ is valid in H: ϕ evaluates to 1 for all interpretations;written H ϕ
The ibd formulas
Let n ≥ 1:
ibd1 := p1 ∨ ¬p1,ibdn+1 := pn+1 ∨ (pn+1 → ibdn).
![Page 75: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/75.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for HA
Corollary
Let H be a nontrivial Heyting algebra and n ≥ 1. TFAE:
1 kdim(H) ≤ n − 1.
2 H ibdn.
3 There does not exist a sequence 1 > b1 > · · · > bn > 0 in Hsuch that bi−1 is dense in Hbi for each i ∈ 1, . . . , n.
4 depth(H∗) ≤ n.
![Page 76: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/76.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for HA
Corollary
Let H be a nontrivial Heyting algebra and n ≥ 1. TFAE:
1 kdim(H) ≤ n − 1.
2 H ibdn.
3 There does not exist a sequence 1 > b1 > · · · > bn > 0 in Hsuch that bi−1 is dense in Hbi for each i ∈ 1, . . . , n.
4 depth(H∗) ≤ n.
![Page 77: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/77.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for HA
Corollary
Let H be a nontrivial Heyting algebra and n ≥ 1. TFAE:
1 kdim(H) ≤ n − 1.
2 H ibdn.
3 There does not exist a sequence 1 > b1 > · · · > bn > 0 in Hsuch that bi−1 is dense in Hbi for each i ∈ 1, . . . , n.
4 depth(H∗) ≤ n.
![Page 78: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/78.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite kdim for HA
Corollary
Let H be a nontrivial Heyting algebra and n ≥ 1. TFAE:
1 kdim(H) ≤ n − 1.
2 H ibdn.
3 There does not exist a sequence 1 > b1 > · · · > bn > 0 in Hsuch that bi−1 is dense in Hbi for each i ∈ 1, . . . , n.
4 depth(H∗) ≤ n.
![Page 79: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/79.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull dimension of a space
Definition
The localic Krull dimension of a topological space X is
ldim(X ) = kdim(Ω(X )) = kdim(℘X ,C)
Corollary: recursive definition of ldim
ldim(X ) = −1 if X = ∅,ldim(X ) ≤ n if ∀D nowhere dense in X , ldim(D) ≤ n − 1,ldim(X ) = n if ldim(X ) ≤ n and ldim(X ) 6≤ n − 1,ldim(X ) =∞ if ldim(X ) 6≤ n for any n = −1, 0, 1, 2, . . . .
![Page 80: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/80.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull dimension of a space
Definition
The localic Krull dimension of a topological space X is
ldim(X ) = kdim(Ω(X )) = kdim(℘X ,C)
Corollary: recursive definition of ldim
ldim(X ) = −1 if X = ∅,ldim(X ) ≤ n if ∀D nowhere dense in X , ldim(D) ≤ n − 1,ldim(X ) = n if ldim(X ) ≤ n and ldim(X ) 6≤ n − 1,ldim(X ) =∞ if ldim(X ) 6≤ n for any n = −1, 0, 1, 2, . . . .
![Page 81: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/81.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull dimension of a space
Definition
The localic Krull dimension of a topological space X is
ldim(X ) = kdim(Ω(X )) = kdim(℘X ,C)
Corollary: recursive definition of ldim
ldim(X ) = −1 if X = ∅,ldim(X ) ≤ n if ∀D nowhere dense in X , ldim(D) ≤ n − 1,ldim(X ) = n if ldim(X ) ≤ n and ldim(X ) 6≤ n − 1,ldim(X ) =∞ if ldim(X ) 6≤ n for any n = −1, 0, 1, 2, . . . .
![Page 82: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/82.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull dimension of a space
Definition
The localic Krull dimension of a topological space X is
ldim(X ) = kdim(Ω(X )) = kdim(℘X ,C)
Corollary: recursive definition of ldim
ldim(X ) = −1 if X = ∅,ldim(X ) ≤ n if ∀D nowhere dense in X , ldim(D) ≤ n − 1,ldim(X ) = n if ldim(X ) ≤ n and ldim(X ) 6≤ n − 1,ldim(X ) =∞ if ldim(X ) 6≤ n for any n = −1, 0, 1, 2, . . . .
![Page 83: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/83.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Localic Krull dimension of a space
Definition
The localic Krull dimension of a topological space X is
ldim(X ) = kdim(Ω(X )) = kdim(℘X ,C)
Corollary: recursive definition of ldim
ldim(X ) = −1 if X = ∅,ldim(X ) ≤ n if ∀D nowhere dense in X , ldim(D) ≤ n − 1,ldim(X ) = n if ldim(X ) ≤ n and ldim(X ) 6≤ n − 1,ldim(X ) =∞ if ldim(X ) 6≤ n for any n = −1, 0, 1, 2, . . . .
![Page 84: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/84.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite localic Krull dimension
Interpretation in X
Via the closure algebra (℘X ,C)Thus ♦ and are C and I resp.ϕ is valid in X : ϕ evaluates to X under all interpretations; writtenX ϕ
Theorem
Let X 6= ∅, n ≥ 1, and Fn+1 be the (n + 1)-element chain. TFAE:
1 ldim(X ) ≤ n − 1.
2 X bdn.
3 There does not exist a sequence E0, . . . ,En of nonemptyclosed subsets of X such that E0 = X and Ei+1 is nowheredense in Ei for each i ∈ 0, . . . , n − 1.
4 Fn+1 is not an interior image of X .
![Page 85: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/85.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite localic Krull dimension
Interpretation in X
Via the closure algebra (℘X ,C)Thus ♦ and are C and I resp.ϕ is valid in X : ϕ evaluates to X under all interpretations; writtenX ϕ
Theorem
Let X 6= ∅, n ≥ 1, and Fn+1 be the (n + 1)-element chain. TFAE:
1 ldim(X ) ≤ n − 1.
2 X bdn.
3 There does not exist a sequence E0, . . . ,En of nonemptyclosed subsets of X such that E0 = X and Ei+1 is nowheredense in Ei for each i ∈ 0, . . . , n − 1.
4 Fn+1 is not an interior image of X .
![Page 86: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/86.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite localic Krull dimension
Interpretation in X
Via the closure algebra (℘X ,C)Thus ♦ and are C and I resp.ϕ is valid in X : ϕ evaluates to X under all interpretations; writtenX ϕ
Theorem
Let X 6= ∅, n ≥ 1, and Fn+1 be the (n + 1)-element chain. TFAE:
1 ldim(X ) ≤ n − 1.
2 X bdn.
3 There does not exist a sequence E0, . . . ,En of nonemptyclosed subsets of X such that E0 = X and Ei+1 is nowheredense in Ei for each i ∈ 0, . . . , n − 1.
4 Fn+1 is not an interior image of X .
![Page 87: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/87.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite localic Krull dimension
Interpretation in X
Via the closure algebra (℘X ,C)Thus ♦ and are C and I resp.ϕ is valid in X : ϕ evaluates to X under all interpretations; writtenX ϕ
Theorem
Let X 6= ∅, n ≥ 1, and Fn+1 be the (n + 1)-element chain. TFAE:
1 ldim(X ) ≤ n − 1.
2 X bdn.
3 There does not exist a sequence E0, . . . ,En of nonemptyclosed subsets of X such that E0 = X and Ei+1 is nowheredense in Ei for each i ∈ 0, . . . , n − 1.
4 Fn+1 is not an interior image of X .
![Page 88: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/88.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Characterizing finite localic Krull dimension
Interpretation in X
Via the closure algebra (℘X ,C)Thus ♦ and are C and I resp.ϕ is valid in X : ϕ evaluates to X under all interpretations; writtenX ϕ
Theorem
Let X 6= ∅, n ≥ 1, and Fn+1 be the (n + 1)-element chain. TFAE:
1 ldim(X ) ≤ n − 1.
2 X bdn.
3 There does not exist a sequence E0, . . . ,En of nonemptyclosed subsets of X such that E0 = X and Ei+1 is nowheredense in Ei for each i ∈ 0, . . . , n − 1.
4 Fn+1 is not an interior image of X .
![Page 89: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/89.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 90: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/90.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 91: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/91.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 92: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/92.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 93: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/93.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 94: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/94.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 95: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/95.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Comparing ldim to other dimension functions
Lemma
Let X be a space
If X is a spectral space, then kdim(X ) ≤ ldim(X ).
If X is a regular space, then ind(X ) ≤ ldim(X ).
If X is a normal space, then Ind(X ) ≤ ldim(X ) anddim(X ) ≤ ldim(X ).
Drawbacks and benefits via some examples
ldim(Rn) = ldim(Q) = ldim(C) =∞ldim(ωn) = n − 1; and so ldim(ωn + 1) = n
ldim(ωω) = ldim(ωω + 1) =∞
![Page 96: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/96.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: I
Lemma
Let X be a T1 space:
ldim(X ) ≤ 0 iff X is discrete
ldim(X ) ≤ 1 iff X is nodec (every nowhere dense set is closed)
Definition
The Zeman formula is zem := ♦p → (p → p)
Esakia et al. 2005
X zem iff X is nodec
![Page 97: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/97.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: I
Lemma
Let X be a T1 space:
ldim(X ) ≤ 0 iff X is discrete
ldim(X ) ≤ 1 iff X is nodec (every nowhere dense set is closed)
Definition
The Zeman formula is zem := ♦p → (p → p)
Esakia et al. 2005
X zem iff X is nodec
![Page 98: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/98.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: I
Lemma
Let X be a T1 space:
ldim(X ) ≤ 0 iff X is discrete
ldim(X ) ≤ 1 iff X is nodec (every nowhere dense set is closed)
Definition
The Zeman formula is zem := ♦p → (p → p)
Esakia et al. 2005
X zem iff X is nodec
![Page 99: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/99.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: I
Lemma
Let X be a T1 space:
ldim(X ) ≤ 0 iff X is discrete
ldim(X ) ≤ 1 iff X is nodec (every nowhere dense set is closed)
Definition
The Zeman formula is zem := ♦p → (p → p)
Esakia et al. 2005
X zem iff X is nodec
![Page 100: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/100.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: II
Definition
The n-Zeman formula iszemn := ( (pn+1 → bdn)→ pn+1)→ (pn+1 → pn+1)
Theorem
Let X be T1. TFAE:
ldim(X ) ≤ n
X zemn
X bdn+1
![Page 101: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/101.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: II
Definition
The n-Zeman formula iszemn := ( (pn+1 → bdn)→ pn+1)→ (pn+1 → pn+1)
Theorem
Let X be T1. TFAE:
ldim(X ) ≤ n
X zemn
X bdn+1
![Page 102: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/102.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
ldim and T1 spaces: II
Definition
The n-Zeman formula iszemn := ( (pn+1 → bdn)→ pn+1)→ (pn+1 → pn+1)
Theorem
Let X be T1. TFAE:
ldim(X ) ≤ n
X zemn
X bdn+1
![Page 103: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/103.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some logics and properties
Definition
For n ≥ 1, put S4n := S4 + bdn and S4.Zn := S4 + zemn
Lemma
S4n+1 ( S4.Zn
S4.Zn has the finite model property
S4.Zn is the logic of uniquely rooted finite frames of depthn + 1
Incompleteness
No logic in [S4n+1,S4.Zn) is complete with respect to a class ofT1 spaces
![Page 104: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/104.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some logics and properties
Definition
For n ≥ 1, put S4n := S4 + bdn and S4.Zn := S4 + zemn
Lemma
S4n+1 ( S4.Zn
S4.Zn has the finite model property
S4.Zn is the logic of uniquely rooted finite frames of depthn + 1
Incompleteness
No logic in [S4n+1,S4.Zn) is complete with respect to a class ofT1 spaces
![Page 105: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/105.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some logics and properties
Definition
For n ≥ 1, put S4n := S4 + bdn and S4.Zn := S4 + zemn
Lemma
S4n+1 ( S4.Zn
S4.Zn has the finite model property
S4.Zn is the logic of uniquely rooted finite frames of depthn + 1
Incompleteness
No logic in [S4n+1,S4.Zn) is complete with respect to a class ofT1 spaces
![Page 106: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/106.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some logics and properties
Definition
For n ≥ 1, put S4n := S4 + bdn and S4.Zn := S4 + zemn
Lemma
S4n+1 ( S4.Zn
S4.Zn has the finite model property
S4.Zn is the logic of uniquely rooted finite frames of depthn + 1
Incompleteness
No logic in [S4n+1,S4.Zn) is complete with respect to a class ofT1 spaces
![Page 107: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/107.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Some logics and properties
Definition
For n ≥ 1, put S4n := S4 + bdn and S4.Zn := S4 + zemn
Lemma
S4n+1 ( S4.Zn
S4.Zn has the finite model property
S4.Zn is the logic of uniquely rooted finite frames of depthn + 1
Incompleteness
No logic in [S4n+1,S4.Zn) is complete with respect to a class ofT1 spaces
![Page 108: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/108.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 109: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/109.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 110: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/110.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 111: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/111.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 112: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/112.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 113: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/113.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 114: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/114.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
The logics S4.Zn
Goal:
Construct a countable crowded Tychonoff space Zn withldim(Zn) = n whose logic is S4.Zn
Ingredients
1 Single frame Bn determining S4.Zn
2 Adjunction spaces (gluing); e.g. wedge sum
3 Cech-Stone compactification and Gleason cover
4 Key ingredient: building block Y , a countable crowdedω-resolvable Tychonoff nodec space such that there is asubspace of βY \ Y that is homeomorphic to βω and for anynowhere dense D ⊆ Y , CD and βω are disjoint.
![Page 115: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/115.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 1: Identify Bn
Bn is the following frame and determines S4.Zn
Bn is obtained using a refined unraveling technique
•HHHH
H
· · ·AA
...
· · ·
AA
...
· · ·
AA
...
· · ·Depth n + 1
Oval =
cluster of
ω points
Forking is ω at each level
![Page 116: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/116.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: base step–build Z1
Choose and fix a point y ∈ YTake topological sum of ω copies of YIdentify each copy of the point y to get Z1
Note B1 is an interior image of Z1
SSSSSSSSS Y Y Y . . .
BBBBBBBBB
•y
Z1
• • •y y y
Y Y Y
. . . -
![Page 117: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/117.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–Chopping Zn
Start with αn : Zn → Bn
Max(Bn) C0 C1 C2
. . .
SSSSSSSS
•αn(y) = r
SSSSSSSSS
•y
Zn Bn-
αn
![Page 118: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/118.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–Chopping Zn
‘Chop’ Zn into Xi = α−1n (R−1Ci )
Max(Bn) C0 C1 C2
. . .
SSSSSSSS
•αn(y) = r
SSSSSSSSS
•y
X0 X1 X2. . .
BBBBBBBBB
BBBBBBBBB
Zn Bn-
αn
![Page 119: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/119.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–general use of Y
For each countable (Tychonoff) space Xi
••••ω
ω∗
Y
Y ∗
βω
βY
![Page 120: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/120.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–general use of Y
There is continuous bijection f : ω → Xi
••••
-f
ω Xiω∗
Y
Y ∗
βω
βY
![Page 121: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/121.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–general use of Y
f continuously extends to g : βω → βXi
••••
-g
-f
ω Xiω∗
Y
Y ∗
βω βXi
βY
![Page 122: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/122.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–general use of Y
Form quotient Q via fibers of g ...
••••
-g
-f
ω Xiω∗
YY Y
Y ∗
βω βXi
βY Q
![Page 123: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/123.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–general use of Y
Form quotient Q via fibers of g ... take subspace Y ∪ Xi
••••
Y
Xiωω∗
Y
Y ∗
βω
βY Q
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Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive case–gluing Y ∪ Xi ’s
Take topological sum of ω copies of Y ∪ Xi
Identify through each copy of Xi to get Ai
Y Y
. . . -
Ai
Y ’s
. . .
AAAAA
AAAAA
Xi Xi
AAAAA
Xi
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Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive step–gluing Ai ’s to get Zn+1
Each Xi is a subset of Zn
Identify the copies of points from Zn
SSSSSSSSS X0 X1 X2
. . .
BBBBBBBBB
Zn+1
Zn
. . .. . .
Y ’s
. . .
Y ’s
. . .
Y ’s
DDDDDDDDD
DDDDDDDDD
Xi Xi+1
Ai Ai+1
. . . . . .
. . .
Y ’s
. . .
Y ’s
-
![Page 126: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/126.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Step 2: Recursive step–gluing Ai ’s to get Zn+1
Each Xi is a subset of Zn
Identify the copies of points from Zn
Send Y ’s ‘above’ Xi to cluster’s above Ci in Bn+1
SSSSSSSSS X0 X1 X2
. . .
BBBBBBBBB
Zn+1
Zn
. . .. . .
Y ’s
. . .
Y ’s
. . .
Y ’s
DDDDDDDDD
DDDDDDDDD
Xi Xi+1
Ai Ai+1
. . . . . .
. . .
Y ’s
. . .
Y ’s
-
![Page 127: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/127.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Conclusions
Theorem
The logic of Zn is S4.Zn
Complete details available at:http://www.illc.uva.nl/Research/Publications/Reports/PP-2016-19.text.pdf
Thank You... Organizers and Audience
Questions ...
![Page 128: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/128.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Conclusions
Theorem
The logic of Zn is S4.Zn
Complete details available at:http://www.illc.uva.nl/Research/Publications/Reports/PP-2016-19.text.pdf
Thank You... Organizers and Audience
Questions ...
![Page 129: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/129.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Conclusions
Theorem
The logic of Zn is S4.Zn
Complete details available at:http://www.illc.uva.nl/Research/Publications/Reports/PP-2016-19.text.pdf
Thank You... Organizers and Audience
Questions ...
![Page 130: Localic Krull Dimensionrmi.tsu.ge/tolo5/Abstracts/Lucero_Bryan.pdf · Point free approach 1 Locale of open subsets )Heyting algebras and intuitionistic logic 2 Power set closure algebra](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e440d67ebd100647909b48e/html5/thumbnails/130.jpg)
Introduction kdim of CA kdim of HA ldim of TOP T1 setting logics S4.Zn
Conclusions
Theorem
The logic of Zn is S4.Zn
Complete details available at:http://www.illc.uva.nl/Research/Publications/Reports/PP-2016-19.text.pdf
Thank You... Organizers and Audience
Questions ...