Dualities in Algebraic Logic

Yde Venema Institute for Logic, Language and Computation Universiteit van Amsterdam https://staff.fnwi.uva.nl/y.venema

13 February 2018 LAC 2018, Melbourne Overview

Introduction Modal Dualities Subdirectly irreducible algebras and rooted structures Vietoris via modal logic Final remarks examples: propositional logic: Boolean algebras intuitionistic logic: Heyting algebras ﬁrst-order logic: cylindric algebras other examples: interpolation: amalgamation completeness: representation abstract algebraic logic: study Logic using methods from (universal) algebra

aim: study logics using methods from (universal) algebra other examples: interpolation: amalgamation completeness: representation abstract algebraic logic: study Logic using methods from (universal) algebra

Algebraic Logic

aim: study logics using methods from (universal) algebra examples: propositional logic: Boolean algebras intuitionistic logic: Heyting algebras ﬁrst-order logic: cylindric algebras interpolation: amalgamation completeness: representation abstract algebraic logic: study Logic using methods from (universal) algebra

Algebraic Logic

aim: study logics using methods from (universal) algebra examples: propositional logic: Boolean algebras intuitionistic logic: Heyting algebras ﬁrst-order logic: cylindric algebras other examples: interpolation: amalgamation completeness: representation abstract algebraic logic: study Logic using methods from (universal) algebra ◦ C and D are dual(ly equivalent) if C and D are equivalent i.e. there are contravariant functors linking C and D

Duality

in mathematics: categorical dualities i.e. there are contravariant functors linking C and D

Duality

in mathematics: categorical dualities ◦ C and D are dual(ly equivalent) if C and D are equivalent Duality

in mathematics: categorical dualities ◦ C and D are dual(ly equivalent) if C and D are equivalent i.e. there are contravariant functors linking C and D algebra geometry

syntax semantics

Stone duality: S j BA Stone Y A

A Fundamental Duality

verbal visual syntax semantics

Stone duality: S j BA Stone Y A

A Fundamental Duality

verbal visual

algebra geometry Stone duality: S j BA Stone Y A

A Fundamental Duality

verbal visual

algebra geometry

syntax semantics A Fundamental Duality

verbal visual

algebra geometry

syntax semantics

Stone duality: S j BA Stone Y A Contravariance In all these examples both categories are concrete!

Variants of Stone duality

Heyting algebra vs Esakia spaces compact regular frames vs compact Hausdorﬀ spaces distributive lattices vs Priestley spaces modal algebras vs topological Kripke structures cylindric algebras vs . . . ... In all these examples both categories are concrete!

Variants of Stone duality

Contravariance Variants of Stone duality

Contravariance In all these examples both categories are concrete! Overview

Introduction Modal Dualities Subdirectly irreducible algebras and rooted structures Vietoris via modal logic Final remarks Overview

Introduction Modal Dualities Subdirectly irreducible algebras and rooted structures Vietoris via modal logic Final remarks topological Kripke structures (TKS) . . . , and of course their morphisms!

Aim: introduce TKS develop duality between MA and TKS

Modal duality

Main characters modal algebras (MA) Kripke structures (KS) Stone spaces (Stone) . . . , and of course their morphisms!

Aim: introduce TKS develop duality between MA and TKS

Modal duality

Main characters modal algebras (MA) Kripke structures (KS) Stone spaces (Stone) topological Kripke structures (TKS) Aim: introduce TKS develop duality between MA and TKS

Modal duality

Main characters modal algebras (MA) Kripke structures (KS) Stone spaces (Stone) topological Kripke structures (TKS) . . . , and of course their morphisms! Modal duality

Main characters modal algebras (MA) Kripke structures (KS) Stone spaces (Stone) topological Kripke structures (TKS) . . . , and of course their morphisms!

Aim: introduce TKS develop duality between MA and TKS 0 h : A → A is an MA-morphism if it preserves all operations: 0 0 0 0 0 0 0 0 I h(a ∨ b ) = h(a ) ∨ h(b ), . . . , h(3 a ) = 3h(a ). MA is the category of modal algebras with MA-morphisms

A modal logic L can be algebraized by a variety VL of modal algebras Modal algebras are (the simplest)BooleanAlgebras withOperators

Modal Algebras

A = (A, ∨, −, ⊥, 3) is a modal algebra if I (A, ∨, −, ⊥) is a Boolean algebra

I 3 : A → A preserves ﬁnite joins: 3⊥ = ⊥ and 3(a ∨ b) = 3a ∨ 3b MA is the category of modal algebras with MA-morphisms

A modal logic L can be algebraized by a variety VL of modal algebras Modal algebras are (the simplest)BooleanAlgebras withOperators

Modal Algebras

A = (A, ∨, −, ⊥, 3) is a modal algebra if I (A, ∨, −, ⊥) is a Boolean algebra

I 3 : A → A preserves ﬁnite joins: 3⊥ = ⊥ and 3(a ∨ b) = 3a ∨ 3b 0 h : A → A is an MA-morphism if it preserves all operations: 0 0 0 0 0 0 0 0 I h(a ∨ b ) = h(a ) ∨ h(b ), . . . , h(3 a ) = 3h(a ). A modal logic L can be algebraized by a variety VL of modal algebras Modal algebras are (the simplest)BooleanAlgebras withOperators

Modal Algebras

A = (A, ∨, −, ⊥, 3) is a modal algebra if I (A, ∨, −, ⊥) is a Boolean algebra

I 3 : A → A preserves ﬁnite joins: 3⊥ = ⊥ and 3(a ∨ b) = 3a ∨ 3b 0 h : A → A is an MA-morphism if it preserves all operations: 0 0 0 0 0 0 0 0 I h(a ∨ b ) = h(a ) ∨ h(b ), . . . , h(3 a ) = 3h(a ). MA is the category of modal algebras with MA-morphisms Modal algebras are (the simplest)BooleanAlgebras withOperators

Modal Algebras

A = (A, ∨, −, ⊥, 3) is a modal algebra if I (A, ∨, −, ⊥) is a Boolean algebra

A modal logic L can be algebraized by a variety VL of modal algebras Modal Algebras

A = (A, ∨, −, ⊥, 3) is a modal algebra if I (A, ∨, −, ⊥) is a Boolean algebra

A modal logic L can be algebraized by a variety VL of modal algebras Modal algebras are (the simplest)BooleanAlgebras withOperators 0 0 f :(S , R ) → (S, R) is a bounded morphism if 0 0 0 0 0 I R s t implies Rf (s )f (t ) 0 0 0 0 0 0 I Rf (s )t implies the existence of t with R s t and f (t ) = t. KS is the category of Kripke structures with bounded morphisms

Kripke structures

A Kripke structure (frame) is a pair S = (S, R) with R ⊆ S × S I these provide the possible-world semantics of modal logic KS is the category of Kripke structures with bounded morphisms

Kripke structures

A Kripke structure (frame) is a pair S = (S, R) with R ⊆ S × S I these provide the possible-world semantics of modal logic 0 0 f :(S , R ) → (S, R) is a bounded morphism if 0 0 0 0 0 I R s t implies Rf (s )f (t ) 0 0 0 0 0 0 I Rf (s )t implies the existence of t with R s t and f (t ) = t. Kripke structures

A (topological) space is a pair (S, τ) where τ is a topology on S A Stone space is a space (S, τ) where τ is I compact,

I Hausdorﬀ

I zero-dimensional (i.e. it has a basis of clopen sets) Stone is the category of Stone spaces and continuous functions From Boolean algebras to Stone spaces:( ·)∗

Objects Given A = (A, ∨, −, ⊥) take A∗ := (Uf (A), σA), where I Uf (A) is the set of ultraﬁlters of A and σ is generated by the basis {a | a ∈ A} I A b I with ab := {u ∈ UF (A) | a ∈ u} 0 0 Arrows Given h : A → A deﬁne h∗ : Uf (A) → Uf (A ) by

0 0 0 h∗(u) := {a ∈ A | ha ∈ u}

Stone duality

From Stone spaces to Boolean algebras:( ·)∗ ∗ Objects Given (S, τ) take( S, τ) := (Clp(τ), ∪, ∼S , ∅) Arrows Given f :(S 0, τ 0) → (S, τ) deﬁne f ∗ : Clp(τ) → Clp(τ 0)

f ∗(X ) := {s0 ∈ S 0 | fs0 ∈ X } I Uf (A) is the set of ultraﬁlters of A and σ is generated by the basis {a | a ∈ A} I A b I with ab := {u ∈ UF (A) | a ∈ u} 0 0 Arrows Given h : A → A deﬁne h∗ : Uf (A) → Uf (A ) by

0 0 0 h∗(u) := {a ∈ A | ha ∈ u}

Stone duality

From Stone spaces to Boolean algebras:( ·)∗ ∗ Objects Given (S, τ) take( S, τ) := (Clp(τ), ∪, ∼S , ∅) Arrows Given f :(S 0, τ 0) → (S, τ) deﬁne f ∗ : Clp(τ) → Clp(τ 0)

f ∗(X ) := {s0 ∈ S 0 | fs0 ∈ X }

From Boolean algebras to Stone spaces:( ·)∗

Objects Given A = (A, ∨, −, ⊥) take A∗ := (Uf (A), σA), where σ is generated by the basis {a | a ∈ A} I A b I with ab := {u ∈ UF (A) | a ∈ u} 0 0 Arrows Given h : A → A deﬁne h∗ : Uf (A) → Uf (A ) by

0 0 0 h∗(u) := {a ∈ A | ha ∈ u}

Stone duality

From Stone spaces to Boolean algebras:( ·)∗ ∗ Objects Given (S, τ) take( S, τ) := (Clp(τ), ∪, ∼S , ∅) Arrows Given f :(S 0, τ 0) → (S, τ) deﬁne f ∗ : Clp(τ) → Clp(τ 0)

f ∗(X ) := {s0 ∈ S 0 | fs0 ∈ X }

From Boolean algebras to Stone spaces:( ·)∗

Objects Given A = (A, ∨, −, ⊥) take A∗ := (Uf (A), σA), where I Uf (A) is the set of ultraﬁlters of A and 0 0 Arrows Given h : A → A deﬁne h∗ : Uf (A) → Uf (A ) by

0 0 0 h∗(u) := {a ∈ A | ha ∈ u}

Stone duality

From Stone spaces to Boolean algebras:( ·)∗ ∗ Objects Given (S, τ) take( S, τ) := (Clp(τ), ∪, ∼S , ∅) Arrows Given f :(S 0, τ 0) → (S, τ) deﬁne f ∗ : Clp(τ) → Clp(τ 0)

f ∗(X ) := {s0 ∈ S 0 | fs0 ∈ X }

From Boolean algebras to Stone spaces:( ·)∗

Objects Given A = (A, ∨, −, ⊥) take A∗ := (Uf (A), σA), where I Uf (A) is the set of ultraﬁlters of A and σ is generated by the basis {a | a ∈ A} I A b I with ab := {u ∈ UF (A) | a ∈ u} Stone duality

From Stone spaces to Boolean algebras:( ·)∗ ∗ Objects Given (S, τ) take( S, τ) := (Clp(τ), ∪, ∼S , ∅) Arrows Given f :(S 0, τ 0) → (S, τ) deﬁne f ∗ : Clp(τ) → Clp(τ 0)

f ∗(X ) := {s0 ∈ S 0 | fs0 ∈ X }

From Boolean algebras to Stone spaces:( ·)∗

0 0 0 h∗(u) := {a ∈ A | ha ∈ u} This is a natural duality evolving around the schizophrenic object 2

Stone duality 2

Theorem ∗ The functors (·) and (·)∗ witness the dual equivalence of BA and Stone. Stone duality 2

Theorem ∗ The functors (·) and (·)∗ witness the dual equivalence of BA and Stone.

This is a natural duality evolving around the schizophrenic object 2 I hRi(X ) := {s ∈ S | R[s] ∩ X 6= ∅} Arrows Given f :(S 0, R0) → (S, R) deﬁne f + as inverse image

The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

From Kripke structures to modal algebras:( ·)+ + Objects Given (S, R) take( S, R) := (PS, ∪, ∼S , ∅, hRi), where The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

From Kripke structures to modal algebras:( ·)+ + Objects Given (S, R) take( S, R) := (PS, ∪, ∼S , ∅, hRi), where I hRi(X ) := {s ∈ S | R[s] ∩ X 6= ∅} Arrows Given f :(S 0, R0) → (S, R) deﬁne f + as inverse image + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

The operation hRi encodes the semantics of the modal diamond Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive (with the opposite functor (·)+ taking the atom structure of a PMA)

Complex algebras

The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS Complex algebras

The operation hRi encodes the semantics of the modal diamond + (S, R) is the complex algebra of (S, R) Complex algebras are perfect modal algebras (PMAs):

I complete, atomic and completely additive + (·) is part of a discrete duality between PMA and KS (with the opposite functor (·)+ taking the atom structure of a PMA) Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

These operations provide a functor: MA → KS A• is the ultraﬁlter structure or canonical structure of A + A embeds in its canonical extension( A•) Open Problem characterize the ultraﬁlter structures modulo isomorphism

Ultraﬁlter structures

From modal algebras to Kripke structures: I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where 0 Arrows Given f : A → A deﬁne f• as inverse image

Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iff ∀a ∈ v.3a ∈ u A• is the ultrafilter structure or canonical structure of A + A embeds in its canonical extension( A•) Open Problem characterize the ultrafilter structures modulo isomorphism

Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

These operations provide a functor: MA → KS + A embeds in its canonical extension( A•) Open Problem characterize the ultraﬁlter structures modulo isomorphism

Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

These operations provide a functor: MA → KS A• is the ultraﬁlter structure or canonical structure of A Open Problem characterize the ultraﬁlter structures modulo isomorphism

Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

These operations provide a functor: MA → KS A• is the ultraﬁlter structure or canonical structure of A + A embeds in its canonical extension( A•) Ultraﬁlter structures

From modal algebras to Kripke structures:

Objects With A = (A, ∨, −, ⊥, 3) take A• := (Uf (A), Q3), where I Q3uv iﬀ ∀a ∈ v.3a ∈ u 0 Arrows Given f : A → A deﬁne f• as inverse image

I R(s) is closed

I TKS is the category with

I objects: topological Kripke structures

I arrows: continuous bounded morphism

Topological Kripke structures

A topological Kripke structure is a triple (S, R, τ) such that I (S, R) is a Kripke structure

I (S, τ) is a Stone space I R(s) is closed

I TKS is the category with

I objects: topological Kripke structures

I arrows: continuous bounded morphism

Topological Kripke structures

A topological Kripke structure is a triple (S, R, τ) such that I (S, R) is a Kripke structure

I (S, τ) is a Stone space

I hRiX is clopen if X ⊆ S is clopen I TKS is the category with

I objects: topological Kripke structures

I arrows: continuous bounded morphism

Topological Kripke structures

A topological Kripke structure is a triple (S, R, τ) such that I (S, R) is a Kripke structure

I (S, τ) is a Stone space

I hRiX is clopen if X ⊆ S is clopen

I R(s) is closed Topological Kripke structures

A topological Kripke structure is a triple (S, R, τ) such that I (S, R) is a Kripke structure

I (S, τ) is a Stone space

I hRiX is clopen if X ⊆ S is clopen

I R(s) is closed

I TKS is the category with

I objects: topological Kripke structures

I arrows: continuous bounded morphism Topological modal duality

From modal algebras to topological Kripke structures:( ·)∗

Objects Given A = (A, ∨, −, ⊥, 3) take A∗ := (Uf (A), Q3, σA) 0 Arrows Given h : A → A deﬁne h∗ as inverse image From topological Kripke structures to modal algebras:( ·)∗ ∗ Objects Given S = (S, R, τ) take S := (Clp(τ), ∪, ∼S , ∅, hRi) Arrows Given f : S0 → S deﬁne f ∗ as inverse image Theorem ∗ The functors (·) and (·)∗ witness the dual equivalence of MA and TKS:

(·)∗ j MA TKS Y ∗ (·) algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? study free modal algebras ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . ...

Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras Remarks

History: J´onsson& Tarski, Lemmon, Scott, Esakia, Goldblatt, . . . algebraic logic || modal logic Research Topics: + (canonicity) Which varieties are closed under (A 7→ (A•) ) + (correspondence) FO properties of S ∼ equational prop’s of S + I e.g. S |= ∀vRvv iﬀ S |= x ≤ 3x (canonicity & correspondence) Sahlqvist theorem (completeness) Which varieties are generated by their PMAs? (completions) canonical extensions, MacNeille completions, . . . study free modal algebras ... Overview

Introduction Modal Dualities Subdirectly irreducible algebras and rooted structures Vietoris via modal logic Final remarks ∼ A is simple if ConA = 2 A is subdirectly irreducible if ConA has a least non-identity element Birkhoﬀ: every variety is generated by its s.i. members

Question What is the dual of an s.i. modal algebra? Folklore Subdirect irreducibility is related to rootedness

Subdirect Irreducibility

Given an algebra A, let ConA be its lattice of congruences Birkhoﬀ: every variety is generated by its s.i. members

Question What is the dual of an s.i. modal algebra? Folklore Subdirect irreducibility is related to rootedness

Subdirect Irreducibility

Given an algebra A, let ConA be its lattice of congruences ∼ A is simple if ConA = 2 A is subdirectly irreducible if ConA has a least non-identity element Question What is the dual of an s.i. modal algebra? Folklore Subdirect irreducibility is related to rootedness

Subdirect Irreducibility

Given an algebra A, let ConA be its lattice of congruences ∼ A is simple if ConA = 2 A is subdirectly irreducible if ConA has a least non-identity element Birkhoﬀ: every variety is generated by its s.i. members Folklore Subdirect irreducibility is related to rootedness

Subdirect Irreducibility

Question What is the dual of an s.i. modal algebra? Subdirect Irreducibility

Question What is the dual of an s.i. modal algebra? Folklore Subdirect irreducibility is related to rootedness R(s) := {t ∈ S | Rst} ω r ∈ S is a root of S if S = R (r)

S is rooted if its collection WS of roots is non-empty

Roots

Auxiliary deﬁnitions ω S n R := n>0 R , 0 n+1 n I where R := IdS and R := R ◦ R ω r ∈ S is a root of S if S = R (r)

S is rooted if its collection WS of roots is non-empty

Roots

Auxiliary deﬁnitions ω S n R := n>0 R , 0 n+1 n I where R := IdS and R := R ◦ R R(s) := {t ∈ S | Rst} Roots

Auxiliary deﬁnitions ω S n R := n>0 R , 0 n+1 n I where R := IdS and R := R ◦ R R(s) := {t ∈ S | Rst} ω r ∈ S is a root of S if S = R (r)

S is rooted if its collection WS of roots is non-empty Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. Proposition (Sambin)

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Proposition (Sambin)

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. (2) If A is s.i. then Int(WA∗ ) 6= ∅ , provided A is (ω-)transitive. Example (Kracht) There are simple algebras of which the dual structure has no roots. Proposition (Rautenberg) A is s.i. iﬀ A∗ has a largest nontrivial, closed hereditary subset.

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. Proposition (Sambin)

(1) If Int(WA∗ ) 6= ∅ then A is s.i. , provided A is (ω-)transitive. Example (Kracht) There are simple algebras of which the dual structure has no roots. Proposition (Rautenberg) A is s.i. iﬀ A∗ has a largest nontrivial, closed hereditary subset.

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. Proposition (Sambin)

(1) If Int(WA∗ ) 6= ∅ then A is s.i. (2) If A is s.i. then Int(WA∗ ) 6= ∅ Example (Kracht) There are simple algebras of which the dual structure has no roots. Proposition (Rautenberg) A is s.i. iﬀ A∗ has a largest nontrivial, closed hereditary subset.

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. Proposition (Sambin)

(1) If Int(WA∗ ) 6= ∅ then A is s.i. (2) If A is s.i. then Int(WA∗ ) 6= ∅ , provided A is (ω-)transitive. Proposition (Rautenberg) A is s.i. iﬀ A∗ has a largest nontrivial, closed hereditary subset.

Subdirect Irreducibility and Rootedness

Proposition (folklore) + WS 6= ∅ (S is rooted) iﬀ S is s.i. Example (Sambin) There are rooted TKSs of which the dual algebra is not s.i. Proposition (Sambin)

(1) If Int(WA∗ ) 6= ∅ then A is s.i. (2) If A is s.i. then Int(WA∗ ) 6= ∅ , provided A is (ω-)transitive. Example (Kracht) There are simple algebras of which the dual structure has no roots. Proposition (Rautenberg) A is s.i. iff A∗ has a largest nontrivial, closed hereditary subset. ω n Q3uv iff ∃n ∈ ω∀a ∈ v. 3 a ∈ u ? Define Q3 by putting ? n Q3uv iff ∀a ∈ v∃n ∈ ω. 3 a ∈ u ? Call r ∈ Uf (A) a topo-root if Q3(r) = Uf (A)

Let TA∗ denote the collection of topo-roots of A∗

Roots and Topo-roots

Fix a modal algebra A. ω r is a root of A∗ iff Q3(r) = Uf (A) ? Define Q3 by putting ? n Q3uv iff ∀a ∈ v∃n ∈ ω. 3 a ∈ u ? Call r ∈ Uf (A) a topo-root if Q3(r) = Uf (A)

Let TA∗ denote the collection of topo-roots of A∗

Roots and Topo-roots

Fix a modal algebra A. ω r is a root of A∗ iﬀ Q3(r) = Uf (A) ω n Q3uv iﬀ ∃n ∈ ω∀a ∈ v. 3 a ∈ u ? Call r ∈ Uf (A) a topo-root if Q3(r) = Uf (A)

Let TA∗ denote the collection of topo-roots of A∗

Roots and Topo-roots

Fix a modal algebra A. ω r is a root of A∗ iff Q3(r) = Uf (A) ω n Q3uv iff ∃n ∈ ω∀a ∈ v. 3 a ∈ u ? Define Q3 by putting ? n Q3uv iff ∀a ∈ v∃n ∈ ω. 3 a ∈ u Let TA∗ denote the collection of topo-roots of A∗

Roots and Topo-roots

Fix a modal algebra A. ω r is a root of A∗ iff Q3(r) = Uf (A) ω n Q3uv iff ∃n ∈ ω∀a ∈ v. 3 a ∈ u ? Define Q3 by putting ? n Q3uv iff ∀a ∈ v∃n ∈ ω. 3 a ∈ u ? Call r ∈ Uf (A) a topo-root if Q3(r) = Uf (A) Roots and Topo-roots

Fix a modal algebra A. ω r is a root of A∗ iff Q3(r) = Uf (A) ω n Q3uv iff ∃n ∈ ω∀a ∈ v. 3 a ∈ u ? Define Q3 by putting ? n Q3uv iff ∀a ∈ v∃n ∈ ω. 3 a ∈ u ? Call r ∈ Uf (A) a topo-root if Q3(r) = Uf (A)

Let TA∗ denote the collection of topo-roots of A∗ Observations

Proposition For any modal algebra A: (1) Q? is transitive (2) Qω ⊆ Q? (3) Q?(u) is hereditary for any ultrafilter u (4) Q?(u) is closed for any ultrafilter u (5) Q?(u) = Qω(u) for any ultrafilter u (6) hQ?i maps opens to opens (7) If Q is transitive then Q = Qω = Q? (2) A is s.i. iff Int(TA∗ ) 6= ∅ Note Earlier results follow from this. Theorem (Birchall) Similar results for distributive modal algebras (based on distr. lattices). Suggestion Develop the modal theory of Q?

Characterizations

Theorem For any modal algebra A: (1) A is simple iﬀ TA∗ = Uf (A) Note Earlier results follow from this. Theorem (Birchall) Similar results for distributive modal algebras (based on distr. lattices). Suggestion Develop the modal theory of Q?

Characterizations

Theorem For any modal algebra A: (1) A is simple iﬀ TA∗ = Uf (A) (2) A is s.i. iﬀ Int(TA∗ ) 6= ∅ Theorem (Birchall) Similar results for distributive modal algebras (based on distr. lattices). Suggestion Develop the modal theory of Q?

Characterizations

Theorem For any modal algebra A: (1) A is simple iﬀ TA∗ = Uf (A) (2) A is s.i. iﬀ Int(TA∗ ) 6= ∅ Note Earlier results follow from this. Suggestion Develop the modal theory of Q?

Characterizations

Theorem For any modal algebra A: (1) A is simple iﬀ TA∗ = Uf (A) (2) A is s.i. iﬀ Int(TA∗ ) 6= ∅ Note Earlier results follow from this. Theorem (Birchall) Similar results for distributive modal algebras (based on distr. lattices). Characterizations

Introduction Modal Dualities Subdirectly irreducible algebras and rooted structures Vietoris via modal logic Final remarks Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)}, I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U: These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), V(X) := hK(X), υτ i is the Vietoris space of X.

The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ The Vietoris construction

Let X = hX , τi be a topological space. K(X) denotes the collection of compact sets With U ⊆ω τ, deﬁne

∇U := {F ∈ K(X) | (F , U) ∈ P(∈)},

where (F , U) ∈ P(∈) if F is ‘properly covered’ by U:

I ∀s ∈ F ∃U ∈ U. s ∈ U and

I ∀U ∈ U∃s ∈ F .s ∈ U These sets ∇U together provide a basis for a topology on K(X), the Vietoris topology υτ V(X) := hK(X), υτ i is the Vietoris space of X. Generate υτ from {h3ia, [3] | a ∈ τ} as a subbasis. Fact The Vietoris construction preserves various properties, including: • compactness • compact Hausdorfness • zero-dimensionality

The Vietoris construction 2

Diﬀerent presentation: For a ∈ τ, deﬁne

3a := {F ∈ K(X) | F ∩ a 6= ∅} 2a := {F ∈ K(X) | F ⊆ a} Fact The Vietoris construction preserves various properties, including: • compactness • compact Hausdorfness • zero-dimensionality

The Vietoris construction 2

Diﬀerent presentation: For a ∈ τ, deﬁne

3a := {F ∈ K(X) | F ∩ a 6= ∅} 2a := {F ∈ K(X) | F ⊆ a}

Generate υτ from {h3ia, [3] | a ∈ τ} as a subbasis. The Vietoris construction 2

Diﬀerent presentation: For a ∈ τ, deﬁne

3a := {F ∈ K(X) | F ∩ a 6= ∅} 2a := {F ∈ K(X) | F ⊆ a}

Generate υτ from {h3ia, [3] | a ∈ τ} as a subbasis. Fact The Vietoris construction preserves various properties, including: • compactness • compact Hausdorfness • zero-dimensionality let Vf : K(X) → P(Y ) be given by Vf (F ) := f [F ] = {fx | x ∈ F }

Then Vf maps compact sets to compact sets. Fact V is a functor on the categories KHaus and Stone.

The Vietoris functor

From now on we restrict to the category KHaus of

I objects: compact Hausdorﬀ spaces

I arrows: continuous maps

Fact Given f : X → Y, Then Vf maps compact sets to compact sets. Fact V is a functor on the categories KHaus and Stone.

The Vietoris functor

From now on we restrict to the category KHaus of

I objects: compact Hausdorﬀ spaces

I arrows: continuous maps

Fact Given f : X → Y, let Vf : K(X) → P(Y ) be given by Vf (F ) := f [F ] = {fx | x ∈ F } Fact V is a functor on the categories KHaus and Stone.

The Vietoris functor

From now on we restrict to the category KHaus of

I objects: compact Hausdorﬀ spaces

I arrows: continuous maps

Fact Given f : X → Y, let Vf : K(X) → P(Y ) be given by Vf (F ) := f [F ] = {fx | x ∈ F }

Then Vf maps compact sets to compact sets. The Vietoris functor

From now on we restrict to the category KHaus of

I objects: compact Hausdorﬀ spaces

I arrows: continuous maps

Fact Given f : X → Y, let Vf : K(X) → P(Y ) be given by Vf (F ) := f [F ] = {fx | x ∈ F }

Then Vf maps compact sets to compact sets. Fact V is a functor on the categories KHaus and Stone. R © ? V

Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) is an arrow R :(S, τ) → V(S, τ) Theorem Topological Kripke frames are Vietoris coalgebras over Stone

Two observations

Observation Stone duality and the Vietoris functor: S j BA Stone Y P R ?

Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) is an arrow R :(S, τ) → V(S, τ) Theorem Topological Kripke frames are Vietoris coalgebras over Stone

Two observations

Observation Stone duality and the Vietoris functor: S j © BA Stone V Y P Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) is an arrow R :(S, τ) → V(S, τ) Theorem Topological Kripke frames are Vietoris coalgebras over Stone

Two observations

Observation Stone duality and the Vietoris functor: S R j © ? BA Stone V Y P is an arrow R :(S, τ) → V(S, τ) Theorem Topological Kripke frames are Vietoris coalgebras over Stone

Two observations

Observation Stone duality and the Vietoris functor: S R j © ? BA Stone V Y P

Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) Theorem Topological Kripke frames are Vietoris coalgebras over Stone

Two observations

Observation Stone duality and the Vietoris functor: S R j © ? BA Stone V Y P

Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) is an arrow R :(S, τ) → V(S, τ) Two observations

Observation Stone duality and the Vietoris functor: S R j © ? BA Stone V Y P

Observation (Esakia) In a TKS (S, R, τ), R : S → P(S) is an arrow R :(S, τ) → V(S, τ) Theorem Topological Kripke frames are Vietoris coalgebras over Stone I It provides a natural framework for notions like

I behavior

I bisimulation/behavioral equivalence

I invariants

I Suﬃciently general to model notions like: input, output, non-determinism, interaction, probability, . . .

Universal Coalgebra

I Universal Coalgebra (Rutten, 2000) is a general mathematical theory for evolving systems I bisimulation/behavioral equivalence

I invariants

I Suﬃciently general to model notions like: input, output, non-determinism, interaction, probability, . . .

Universal Coalgebra

I Universal Coalgebra (Rutten, 2000) is a general mathematical theory for evolving systems

I It provides a natural framework for notions like

I behavior I invariants

I Suﬃciently general to model notions like: input, output, non-determinism, interaction, probability, . . .

Universal Coalgebra

I Universal Coalgebra (Rutten, 2000) is a general mathematical theory for evolving systems

I It provides a natural framework for notions like

I behavior

I bisimulation/behavioral equivalence I Suﬃciently general to model notions like: input, output, non-determinism, interaction, probability, . . .

Universal Coalgebra

I Universal Coalgebra (Rutten, 2000) is a general mathematical theory for evolving systems

I It provides a natural framework for notions like

I behavior

I bisimulation/behavioral equivalence

I invariants Universal Coalgebra

I Universal Coalgebra (Rutten, 2000) is a general mathematical theory for evolving systems

I It provides a natural framework for notions like

I behavior

I bisimulation/behavioral equivalence

I invariants

I Suﬃciently general to model notions like: input, output, non-determinism, interaction, probability, . . . An T-coalgebra is a pair (c, γ : c → Tc). 0 0 A coalgebra morphism between two coalgebras (c , γ ) and (c, γ) is an arrow f : c0 → C with f c0 - c

γ0 γ ? Tf ? Tc0 - Tc

Examples: Kripke structures are P-coalgebras over Set deterministics ﬁnite automata are coalgebras over Set

Coalgebras and their morphisms

Let T : C → C be an endofunctor on the category C 0 0 A coalgebra morphism between two coalgebras (c , γ ) and (c, γ) is an arrow f : c0 → C with f c0 - c

γ0 γ ? Tf ? Tc0 - Tc

Examples: Kripke structures are P-coalgebras over Set deterministics ﬁnite automata are coalgebras over Set

Coalgebras and their morphisms

Let T : C → C be an endofunctor on the category C An T-coalgebra is a pair (c, γ : c → Tc). Examples: Kripke structures are P-coalgebras over Set deterministics ﬁnite automata are coalgebras over Set

Coalgebras and their morphisms

Let T : C → C be an endofunctor on the category C An T-coalgebra is a pair (c, γ : c → Tc). 0 0 A coalgebra morphism between two coalgebras (c , γ ) and (c, γ) is an arrow f : c0 → C with f c0 - c

γ0 γ ? Tf ? Tc0 - Tc Coalgebras and their morphisms

γ0 γ ? Tf ? Tc0 - Tc

Examples: Kripke structures are P-coalgebras over Set deterministics ﬁnite automata are coalgebras over Set Manifestations: The ﬁnal V-coalgebra ∼ the canonical general frame (C, R, τ), the map s 7→ R(s) is a homeomorphism R :(C, τ) → V(C, τ) Duality: S R j © ? BA Stone V Y P

Vietoris coalgebras

∼ Theorem TKS = CoalgV(Stone) Manifestations: The ﬁnal V-coalgebra ∼ the canonical general frame (C, R, τ), Duality: S R j © ? BA Stone V Y P

Vietoris coalgebras

∼ Theorem TKS = CoalgV(Stone) Manifestations: The ﬁnal V-coalgebra ∼ the canonical general frame (C, R, τ), the map s 7→ R(s) is a homeomorphism R :(C, τ) → V(C, τ) ?

Vietoris coalgebras

∼ Theorem TKS = CoalgV(Stone) Manifestations: The ﬁnal V-coalgebra ∼ the canonical general frame (C, R, τ), the map s 7→ R(s) is a homeomorphism R :(C, τ) → V(C, τ) Duality: S R j © BA Stone V Y P Vietoris coalgebras

∼ Theorem TKS = CoalgV(Stone) Manifestations: The ﬁnal V-coalgebra ∼ the canonical general frame (C, R, τ), the map s 7→ R(s) is a homeomorphism R :(C, τ) → V(C, τ) Duality: S R j © ? BA Stone V Y P M

Johnstone: describe M via generators and relations Given a BA B,M B is the Boolean algebra I generated by the set {3b : b ∈ B}

I modulo the relations 3(a ∨ b) = 3a ∨ 3b and 3> = >

∼ Theorem (Kupke, Kurz & Venema) ModAlg = ALgBA(M).

The topological modal duality is an algebra|coalgebra duality

Modal Logic Dualizes the Vietoris Functor

I modulo the relations 3(a ∨ b) = 3a ∨ 3b and 3> = >

∼ Theorem (Kupke, Kurz & Venema) ModAlg = ALgBA(M).

The topological modal duality is an algebra|coalgebra duality

Modal Logic Dualizes the Vietoris Functor

Johnstone: describe M via generators and relations ∼ Theorem (Kupke, Kurz & Venema) ModAlg = ALgBA(M).

The topological modal duality is an algebra|coalgebra duality

Modal Logic Dualizes the Vietoris Functor

Johnstone: describe M via generators and relations Given a BA B,M B is the Boolean algebra I generated by the set {3b : b ∈ B}

I modulo the relations 3(a ∨ b) = 3a ∨ 3b and 3> = > The topological modal duality is an algebra|coalgebra duality

Modal Logic Dualizes the Vietoris Functor

Johnstone: describe M via generators and relations Given a BA B,M B is the Boolean algebra I generated by the set {3b : b ∈ B}

I modulo the relations 3(a ∨ b) = 3a ∨ 3b and 3> = >

∼ Theorem (Kupke, Kurz & Venema) ModAlg = ALgBA(M). Modal Logic Dualizes the Vietoris Functor

Johnstone: describe M via generators and relations Given a BA B,M B is the Boolean algebra I generated by the set {3b : b ∈ B}

I modulo the relations 3(a ∨ b) = 3a ∨ 3b and 3> = >

∼ Theorem (Kupke, Kurz & Venema) ModAlg = ALgBA(M).

Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone)

Variation: Pointfree Topology

Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone)

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. S j KRFr KHaus Y P R M

Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone)

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. S j © KRFr KHaus V Y P Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone)

Variation: Pointfree Topology

Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone) Vietoris pointfree (Johnstone Functor)

Given a frame L, deﬁne L2 := {2a | a ∈ L} and L3 := {3a | a ∈ L}.

V V ML := FrhL2 ] L3 | 2( A) = 2a (A ∈ PωL) W Wa∈A 3( A) = a∈A 3a (A ∈ PωL)

2a ∧ 3b ≤ 3(a ∧ b) 2(a ∨ b) ≤ 2a ∨ 3b

2(F A) = F 2a (A ∈ PL directed) F Fa∈A 3( A) = a∈A 3a (A ∈ PL directed)

i I Now think of ∇ as a primitive modality I This modality has many manifestations in modal logic

I normal forms (Fine)

I coalgebraic modal logic (Moss)

I automata theory (Walukiewicz)

I May develop ∇-logic...

I . . . and formulate the functor M accordingly, in terms of ∇

Vietoris and the Cover Modality ∇

I Vietoris used the ∇-constructor on Pωτ I This modality has many manifestations in modal logic

I normal forms (Fine)

I coalgebraic modal logic (Moss)

I automata theory (Walukiewicz)

I May develop ∇-logic...

I . . . and formulate the functor M accordingly, in terms of ∇

Vietoris and the Cover Modality ∇

I Vietoris used the ∇-constructor on Pωτ

I Now think of ∇ as a primitive modality I May develop ∇-logic...

I . . . and formulate the functor M accordingly, in terms of ∇

Vietoris and the Cover Modality ∇

I Vietoris used the ∇-constructor on Pωτ

I Now think of ∇ as a primitive modality I This modality has many manifestations in modal logic

I normal forms (Fine)

I coalgebraic modal logic (Moss)

I automata theory (Walukiewicz) I . . . and formulate the functor M accordingly, in terms of ∇

Vietoris and the Cover Modality ∇

I Vietoris used the ∇-constructor on Pωτ

I Now think of ∇ as a primitive modality I This modality has many manifestations in modal logic

I normal forms (Fine)

I coalgebraic modal logic (Moss)

I automata theory (Walukiewicz)

I May develop ∇-logic... Vietoris and the Cover Modality ∇

I Vietoris used the ∇-constructor on Pωτ

I Now think of ∇ as a primitive modality I This modality has many manifestations in modal logic

I normal forms (Fine)

I coalgebraic modal logic (Moss)

I automata theory (Walukiewicz)

I May develop ∇-logic...

I . . . and formulate the functor M accordingly, in terms of ∇ Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL)

New directions Deﬁne the T-powerlocale of a frame L as

New directions

Fix a standard set functor T that preserves weak pullbacks. where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL)

New directions

Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, (∇1) ∇α ≤ ∇β (α T≤ β) ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL)

New directions

Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, where the relations are as follows: ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL)

New directions

Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL)

New directions

Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ New directions

Fix a standard set functor T that preserves weak pullbacks.

Deﬁne the T-powerlocale of a frame L as

MTL := FrhTωL | (∇1), (∇2), (∇3)i, where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) ^ _ V (∇2) ∇γ ≤ {∇(T )Ψ | Ψ ∈ SRD(Γ)} (Γ ∈ PωTωL) γ∈Γ W _ (∇3) ∇(T )Φ ≤ {∇β | β T∈ Φ} (Φ ∈ TωPL) Theorem (V., Vickers & Vosmaer) Given a set funtor T that preserves weak pullbacks:

MT provides a functor on the category Fr of frames. ∼ MT generalizes Johnstone’s M: M = MP. MT preserves regularity, zero-dimensionality, and Stone-ness. MT restricts to a functor on KRFr (compact regular frames) provided T preserves ﬁniteness

Describe the dual of MT for an arbitrary set functor T!

Some results ∼ MT generalizes Johnstone’s M: M = MP. MT preserves regularity, zero-dimensionality, and Stone-ness. MT restricts to a functor on KRFr (compact regular frames) provided T preserves ﬁniteness

Describe the dual of MT for an arbitrary set functor T!

Some results