Introduction Varieties not closed under completions Overview of completions Conclusions

Varieties (not) closed under completionsKyoto 2006

Tadeusz Litak

School of Information Science, JAIST

June 1, 2006

Introduction Varieties not closed under completions Overview of completions Conclusions

1 IntroductionLatticesComplete additivity and residuationFLew-algebrasBAOs and diagonalizable algebrasComplete generation and closure under completions

2 Varieties not closed under completionsLob algebrasMV-algebras

3 Overview of completionsDedekind-MacNeille completionsCanonical completions and beyond

4 Conclusions

Introduction Varieties not closed under completions Overview of completions Conclusions

A lattice

I assume the definition is known, but just in case: a posetwhere every pair {x , y} has l.u.b x ∨ y and g.l.b x ∧ yHowever, we prefer algebraic definition here: algebra 〈L,∧,∨〉,where

a ∧ a = a

a ∧ (b ∧ c) = (a ∧ b) ∧ c

a ∧ b = b ∧ a

a ∧ (a ∨ b) = a

. . . and the same with ∧ and ∨ replacedMost of the time, we will be interested in distributive lattices(a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c))

Introduction Varieties not closed under completions Overview of completions Conclusions

A lattice

I assume the definition is known, but just in case: a posetwhere every pair {x , y} has l.u.b x ∨ y and g.l.b x ∧ yHowever, we prefer algebraic definition here: algebra 〈L,∧,∨〉,where

a ∧ a = a

a ∧ (b ∧ c) = (a ∧ b) ∧ c

a ∧ b = b ∧ a

a ∧ (a ∨ b) = a

. . . and the same with ∧ and ∨ replacedMost of the time, we will be interested in distributive lattices(a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c))

Introduction Varieties not closed under completions Overview of completions Conclusions

A lattice

I assume the definition is known, but just in case: a posetwhere every pair {x , y} has l.u.b x ∨ y and g.l.b x ∧ yHowever, we prefer algebraic definition here: algebra 〈L,∧,∨〉,where

a ∧ a = a

a ∧ (b ∧ c) = (a ∧ b) ∧ c

a ∧ b = b ∧ a

a ∧ (a ∨ b) = a

. . . and the same with ∧ and ∨ replacedMost of the time, we will be interested in distributive lattices(a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c))

Introduction Varieties not closed under completions Overview of completions Conclusions

A lattice

I assume the definition is known, but just in case: a posetwhere every pair {x , y} has l.u.b x ∨ y and g.l.b x ∧ yHowever, we prefer algebraic definition here: algebra 〈L,∧,∨〉,where

a ∧ a = a

a ∧ (b ∧ c) = (a ∧ b) ∧ c

a ∧ b = b ∧ a

a ∧ (a ∨ b) = a

. . . and the same with ∧ and ∨ replacedMost of the time, we will be interested in distributive lattices(a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c))

Introduction Varieties not closed under completions Overview of completions Conclusions

A lattice

I assume the definition is known, but just in case: a posetwhere every pair {x , y} has l.u.b x ∨ y and g.l.b x ∧ yHowever, we prefer algebraic definition here: algebra 〈L,∧,∨〉,where

a ∧ a = a

a ∧ (b ∧ c) = (a ∧ b) ∧ c

a ∧ b = b ∧ a

a ∧ (a ∨ b) = a

. . . and the same with ∧ and ∨ replacedMost of the time, we will be interested in distributive lattices(a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c))

Introduction Varieties not closed under completions Overview of completions Conclusions

Complete lattices

We will be particularly interested in complete lattices: whereevery set X has g.l.b.

∧X and l.u.b.

∨X

Recall that every complete lattice L is bounded: there is agreatest element

∨L = > and smallest element

∧L = ⊥

We can weaken the notion of completeness to, e.g.,ω-completeness: every countable set has g.l.b. and l.u.b.

Introduction Varieties not closed under completions Overview of completions Conclusions

Complete lattices

We will be particularly interested in complete lattices: whereevery set X has g.l.b.

∧X and l.u.b.

∨X

Recall that every complete lattice L is bounded: there is agreatest element

∨L = > and smallest element

∧L = ⊥

We can weaken the notion of completeness to, e.g.,ω-completeness: every countable set has g.l.b. and l.u.b.

Introduction Varieties not closed under completions Overview of completions Conclusions

Complete lattices

We will be particularly interested in complete lattices: whereevery set X has g.l.b.

∧X and l.u.b.

∨X

Recall that every complete lattice L is bounded: there is agreatest element

∨L = > and smallest element

∧L = ⊥

We can weaken the notion of completeness to, e.g.,ω-completeness: every countable set has g.l.b. and l.u.b.

Introduction Varieties not closed under completions Overview of completions Conclusions

Other lattice-theoretical notions

Definition

x is join-irreducible (join-prime) if x = y ∨ z (x ≤ y ∨ z) impliesx = y or x = z (x ≤ y or x ≤ z).

In distributive lattices bothnotions coincide. Definitions for meets are dual. Completelyjoin-irreducible (join-prime) elements are indecomposable byinfinite joins. An atom is a minimal element distinct from thebottom. Every atom is completely join-prime. A lattice is atomicif every element is above an atomic and atomistic if everyelement is a join of atoms. In atomistic lattices, the completelyirreducible elements are exactly the atoms

We haven’t defined boolean algebras yet, but I guess youknown: in this case, being atomic means being atomistic. Also,atoms are the only join-prime elements

Introduction Varieties not closed under completions Overview of completions Conclusions

Other lattice-theoretical notions

Definition

x is join-irreducible (join-prime) if x = y ∨ z (x ≤ y ∨ z) impliesx = y or x = z (x ≤ y or x ≤ z). In distributive lattices bothnotions coincide. Definitions for meets are dual.

Completelyjoin-irreducible (join-prime) elements are indecomposable byinfinite joins. An atom is a minimal element distinct from thebottom. Every atom is completely join-prime. A lattice is atomicif every element is above an atomic and atomistic if everyelement is a join of atoms. In atomistic lattices, the completelyirreducible elements are exactly the atoms

We haven’t defined boolean algebras yet, but I guess youknown: in this case, being atomic means being atomistic. Also,atoms are the only join-prime elements

Introduction Varieties not closed under completions Overview of completions Conclusions

Other lattice-theoretical notions

Definition

x is join-irreducible (join-prime) if x = y ∨ z (x ≤ y ∨ z) impliesx = y or x = z (x ≤ y or x ≤ z). In distributive lattices bothnotions coincide. Definitions for meets are dual. Completelyjoin-irreducible (join-prime) elements are indecomposable byinfinite joins.

An atom is a minimal element distinct from thebottom. Every atom is completely join-prime. A lattice is atomicif every element is above an atomic and atomistic if everyelement is a join of atoms. In atomistic lattices, the completelyirreducible elements are exactly the atoms

We haven’t defined boolean algebras yet, but I guess youknown: in this case, being atomic means being atomistic. Also,atoms are the only join-prime elements

Introduction Varieties not closed under completions Overview of completions Conclusions

Other lattice-theoretical notions

Definition

x is join-irreducible (join-prime) if x = y ∨ z (x ≤ y ∨ z) impliesx = y or x = z (x ≤ y or x ≤ z). In distributive lattices bothnotions coincide. Definitions for meets are dual. Completelyjoin-irreducible (join-prime) elements are indecomposable byinfinite joins. An atom is a minimal element distinct from thebottom. Every atom is completely join-prime. A lattice is atomicif every element is above an atomic and atomistic if everyelement is a join of atoms. In atomistic lattices, the completelyirreducible elements are exactly the atoms

We haven’t defined boolean algebras yet, but I guess youknown: in this case, being atomic means being atomistic. Also,atoms are the only join-prime elements

Introduction Varieties not closed under completions Overview of completions Conclusions

Other lattice-theoretical notions

Definition

x is join-irreducible (join-prime) if x = y ∨ z (x ≤ y ∨ z) impliesx = y or x = z (x ≤ y or x ≤ z). In distributive lattices bothnotions coincide. Definitions for meets are dual. Completelyjoin-irreducible (join-prime) elements are indecomposable byinfinite joins. An atom is a minimal element distinct from thebottom. Every atom is completely join-prime. A lattice is atomicif every element is above an atomic and atomistic if everyelement is a join of atoms. In atomistic lattices, the completelyirreducible elements are exactly the atoms

We haven’t defined boolean algebras yet, but I guess youknown: in this case, being atomic means being atomistic. Also,atoms are the only join-prime elements

Introduction Varieties not closed under completions Overview of completions Conclusions

Filters and ideals

A filter: any non-empty subset F of a lattice s.t. a ∧ b ∈ S iffa ∈ S and b ∈ S

An ideal: dual filter. A non-empty subset I s.t. a ∨ b ∈ S iffa ∈ S and b ∈ SA prime filter: A filter F s.t. for every a,b, a ∨ b ∈ F iff a ∈ F orb ∈ F . Prime ideal defined duallyThe principal filter generated by a: [a) = {b|a ∧ b = a}.Principal ideal (a] defined dually. In a distributive lattice,principal filters of join-prime elements are prime

Introduction Varieties not closed under completions Overview of completions Conclusions

Filters and ideals

A filter: any non-empty subset F of a lattice s.t. a ∧ b ∈ S iffa ∈ S and b ∈ SAn ideal: dual filter. A non-empty subset I s.t. a ∨ b ∈ S iffa ∈ S and b ∈ S

A prime filter: A filter F s.t. for every a,b, a ∨ b ∈ F iff a ∈ F orb ∈ F . Prime ideal defined duallyThe principal filter generated by a: [a) = {b|a ∧ b = a}.Principal ideal (a] defined dually. In a distributive lattice,principal filters of join-prime elements are prime

Introduction Varieties not closed under completions Overview of completions Conclusions

Filters and ideals

A filter: any non-empty subset F of a lattice s.t. a ∧ b ∈ S iffa ∈ S and b ∈ SAn ideal: dual filter. A non-empty subset I s.t. a ∨ b ∈ S iffa ∈ S and b ∈ SA prime filter: A filter F s.t. for every a,b, a ∨ b ∈ F iff a ∈ F orb ∈ F . Prime ideal defined dually

The principal filter generated by a: [a) = {b|a ∧ b = a}.Principal ideal (a] defined dually. In a distributive lattice,principal filters of join-prime elements are prime

Introduction Varieties not closed under completions Overview of completions Conclusions

Filters and ideals

A filter: any non-empty subset F of a lattice s.t. a ∧ b ∈ S iffa ∈ S and b ∈ SAn ideal: dual filter. A non-empty subset I s.t. a ∨ b ∈ S iffa ∈ S and b ∈ SA prime filter: A filter F s.t. for every a,b, a ∨ b ∈ F iff a ∈ F orb ∈ F . Prime ideal defined duallyThe principal filter generated by a: [a) = {b|a ∧ b = a}.Principal ideal (a] defined dually. In a distributive lattice,principal filters of join-prime elements are prime

Introduction Varieties not closed under completions Overview of completions Conclusions

Complete additivity and residuals - unary case

f : L 7→ L is completely additive if for every {xi}i∈I which has ag.l.b., f (

∨xi) =

∨f (xi)

f : L 7→ L is called residuated if there is a map f← : L 7→ L s.t.f (x) ≤ y iff x ≤ f←(y)

Theorem (Jonsson and Tarski? Blyth and Janowitz?)

Every residuated operation is completely additive

f (p) =∧{q | p ≤ f←(q)}

f←(q) =∨{p | f (p) ≤ q}

Introduction Varieties not closed under completions Overview of completions Conclusions

Residuation — binary case

We say that · : L× L 7→ L is residuated if it is residuatedcomponentwise: there are maps \ : L× L 7→ L and/ : L× L 7→ L s.t.

x · y ≤ z ⇔ y ≤ x \ z ⇔ x ≤ z/y

If · is commutative, it boils down to existence of →: L× L 7→ L

s.t.

x · y ≤ z ⇔ y ≤ x → z ⇔ x ≤ y → z

If · is a monoid operation with unit 1, we say 〈L,∧,∨, ·,→,1〉 is a(commutative) residuated lattice. They all satisfy

x ·∨

yi =∨

x · yi , (∨

xi) · y =∨

(xi · y)

Introduction Varieties not closed under completions Overview of completions Conclusions

Residuation — binary case

We say that · : L× L 7→ L is residuated if it is residuatedcomponentwise: there are maps \ : L× L 7→ L and/ : L× L 7→ L s.t.

x · y ≤ z ⇔ y ≤ x \ z ⇔ x ≤ z/y

If · is commutative, it boils down to existence of →: L× L 7→ L

s.t.

x · y ≤ z ⇔ y ≤ x → z ⇔ x ≤ y → z

If · is a monoid operation with unit 1, we say 〈L,∧,∨, ·,→,1〉 is a(commutative) residuated lattice. They all satisfy

x ·∨

yi =∨

x · yi , (∨

xi) · y =∨

(xi · y)

Introduction Varieties not closed under completions Overview of completions Conclusions

Residuation — binary case

We say that · : L× L 7→ L is residuated if it is residuatedcomponentwise: there are maps \ : L× L 7→ L and/ : L× L 7→ L s.t.

x · y ≤ z ⇔ y ≤ x \ z ⇔ x ≤ z/y

If · is commutative, it boils down to existence of →: L× L 7→ L

s.t.

x · y ≤ z ⇔ y ≤ x → z ⇔ x ≤ y → z

If · is a monoid operation with unit 1, we say 〈L,∧,∨, ·,→,1〉 is a(commutative) residuated lattice. They all satisfy

x ·∨

yi =∨

x · yi , (∨

xi) · y =∨

(xi · y)

Introduction Varieties not closed under completions Overview of completions Conclusions

`-groups

If · is a (commutative) group operation with x−1 = x → 1, wesay 〈L,∧,∨, ·,→,1〉 is a (commutative) `-group

An important property of an `-groups: they are duallyresiduated. If you reverse `-group upside down, you will still getan `-group. Hence, they satisfy

x ·∧

yi =∧

(x · yi)

It does not hold in general in residuated lattices. Another niceand exotic property of `-groups is that the following holds

x ∨∧i∈I

yi =∧i∈I

(x ∨ yi)

Introduction Varieties not closed under completions Overview of completions Conclusions

`-groups

If · is a (commutative) group operation with x−1 = x → 1, wesay 〈L,∧,∨, ·,→,1〉 is a (commutative) `-groupAn important property of an `-groups: they are duallyresiduated. If you reverse `-group upside down, you will still getan `-group. Hence, they satisfy

x ·∧

yi =∧

(x · yi)

It does not hold in general in residuated lattices.

Another niceand exotic property of `-groups is that the following holds

x ∨∧i∈I

yi =∧i∈I

(x ∨ yi)

Introduction Varieties not closed under completions Overview of completions Conclusions

`-groups

If · is a (commutative) group operation with x−1 = x → 1, wesay 〈L,∧,∨, ·,→,1〉 is a (commutative) `-groupAn important property of an `-groups: they are duallyresiduated. If you reverse `-group upside down, you will still getan `-group. Hence, they satisfy

x ·∧

yi =∧

(x · yi)

It does not hold in general in residuated lattices. Another niceand exotic property of `-groups is that the following holds

x ∨∧i∈I

yi =∧i∈I

(x ∨ yi)

Introduction Varieties not closed under completions Overview of completions Conclusions

FLew-algebras

`-groups are interesting algebraically but of no direct relation tologic. We will focus our attention on a different kind ofstructures:

Definition

〈L,∧,∨, ·,→,>,⊥〉 is a FLew-algebra if 〈L,∧,∨,>,⊥〉 is abounded lattice and 〈L,∧,∨, ·,→,>, 〉 is a commutativeresiduated lattice. That is, monoid unit is the top of the lattice

All FLew-algebras satisfy x · y ≤ x ∧ y . A convention:¬x := x → ⊥

Introduction Varieties not closed under completions Overview of completions Conclusions

FLew-algebras

`-groups are interesting algebraically but of no direct relation tologic. We will focus our attention on a different kind ofstructures:

Definition

〈L,∧,∨, ·,→,>,⊥〉 is a FLew-algebra if 〈L,∧,∨,>,⊥〉 is abounded lattice and 〈L,∧,∨, ·,→,>, 〉 is a commutativeresiduated lattice. That is, monoid unit is the top of the lattice

All FLew-algebras satisfy x · y ≤ x ∧ y . A convention:¬x := x → ⊥

Introduction Varieties not closed under completions Overview of completions Conclusions

FLew-algebras

`-groups are interesting algebraically but of no direct relation tologic. We will focus our attention on a different kind ofstructures:

Definition

〈L,∧,∨, ·,→,>,⊥〉 is a FLew-algebra if 〈L,∧,∨,>,⊥〉 is abounded lattice and 〈L,∧,∨, ·,→,>, 〉 is a commutativeresiduated lattice. That is, monoid unit is the top of the lattice

All FLew-algebras satisfy x · y ≤ x ∧ y . A convention:¬x := x → ⊥

Introduction Varieties not closed under completions Overview of completions Conclusions

MV -algebras and Heyting algebras

Most interesting examples of FLew-algebras:

Definition

A MV-algebra: satisfies x ∨ y = (x → y) → y

Definition

A Heyting algebra: satisfies x · y = x ∧ y

Definition

A boolean algebra: a Heyting algebra which satisfiesx ∨ ¬x = >

All these algebras are distributive and hence have a niceset-theoretical representation. Of course, representation theoryworks best for boolean algebras. We will return to the subject

Introduction Varieties not closed under completions Overview of completions Conclusions

MV -algebras and Heyting algebras

Most interesting examples of FLew-algebras:

Definition

A MV-algebra: satisfies x ∨ y = (x → y) → y

Definition

A Heyting algebra: satisfies x · y = x ∧ y

Definition

A boolean algebra: a Heyting algebra which satisfiesx ∨ ¬x = >

All these algebras are distributive and hence have a niceset-theoretical representation. Of course, representation theoryworks best for boolean algebras. We will return to the subject

Introduction Varieties not closed under completions Overview of completions Conclusions

MV -algebras and Heyting algebras

Most interesting examples of FLew-algebras:

Definition

A MV-algebra: satisfies x ∨ y = (x → y) → y

Definition

A Heyting algebra: satisfies x · y = x ∧ y

Definition

A boolean algebra: a Heyting algebra which satisfiesx ∨ ¬x = >

All these algebras are distributive and hence have a niceset-theoretical representation. Of course, representation theoryworks best for boolean algebras. We will return to the subject

Introduction Varieties not closed under completions Overview of completions Conclusions

MV -algebras and Heyting algebras

Most interesting examples of FLew-algebras:

Definition

A MV-algebra: satisfies x ∨ y = (x → y) → y

Definition

A Heyting algebra: satisfies x · y = x ∧ y

Definition

A boolean algebra: a Heyting algebra which satisfiesx ∨ ¬x = >

All these algebras are distributive and hence have a niceset-theoretical representation. Of course, representation theoryworks best for boolean algebras. We will return to the subject

Introduction Varieties not closed under completions Overview of completions Conclusions

BAOs

We are most interested in BAOs: boolean algebras with (unary)operator(s) ♦: operations satisfying ♦(x ∨ y) = ♦x ∨ ♦y and♦⊥ = ⊥

Notation: �x := ¬♦¬xOperators are additive, but they do not need to be completelyadditive. However, we will see now an important example ofones which are

Introduction Varieties not closed under completions Overview of completions Conclusions

BAOs

We are most interested in BAOs: boolean algebras with (unary)operator(s) ♦: operations satisfying ♦(x ∨ y) = ♦x ∨ ♦y and♦⊥ = ⊥Notation: �x := ¬♦¬x

Operators are additive, but they do not need to be completelyadditive. However, we will see now an important example ofones which are

Introduction Varieties not closed under completions Overview of completions Conclusions

BAOs

We are most interested in BAOs: boolean algebras with (unary)operator(s) ♦: operations satisfying ♦(x ∨ y) = ♦x ∨ ♦y and♦⊥ = ⊥Notation: �x := ¬♦¬xOperators are additive, but they do not need to be completelyadditive. However, we will see now an important example ofones which are

Introduction Varieties not closed under completions Overview of completions Conclusions

Complex algebras

Let F := 〈W , {Ri}i∈I〉 be a relational structure — we assume allrelations are binary. Then the powerset algebra of F withoperators ♦iX := {y ∈ W | ∃x ∈ X .yRix} is a BAO withcompletely additive operators. Moreover, it’s also complete andatomic. We denote it by F+

These operators are not only completely additive, but in factresiduated:♦←i X = �P

i X := {y ∈ W | ∀x ∈ W .(xRiy ⇒ x ∈ X )}. In otherwords, residuals are duals of conjugate operatorsSometimes, we can choose to add residuals/conjugatesexplicitly to the signature. That is, for every operator ♦i add anew operator ♦P

i satisfying

x ≤ �i♦Pi x

x ≤ �Pi ♦ix

Introduction Varieties not closed under completions Overview of completions Conclusions

Complex algebras

Let F := 〈W , {Ri}i∈I〉 be a relational structure — we assume allrelations are binary. Then the powerset algebra of F withoperators ♦iX := {y ∈ W | ∃x ∈ X .yRix} is a BAO withcompletely additive operators. Moreover, it’s also complete andatomic. We denote it by F+

These operators are not only completely additive, but in factresiduated:♦←i X = �P

i X := {y ∈ W | ∀x ∈ W .(xRiy ⇒ x ∈ X )}. In otherwords, residuals are duals of conjugate operators

Sometimes, we can choose to add residuals/conjugatesexplicitly to the signature. That is, for every operator ♦i add anew operator ♦P

i satisfying

x ≤ �i♦Pi x

x ≤ �Pi ♦ix

Introduction Varieties not closed under completions Overview of completions Conclusions

Complex algebras

Let F := 〈W , {Ri}i∈I〉 be a relational structure — we assume allrelations are binary. Then the powerset algebra of F withoperators ♦iX := {y ∈ W | ∃x ∈ X .yRix} is a BAO withcompletely additive operators. Moreover, it’s also complete andatomic. We denote it by F+

These operators are not only completely additive, but in factresiduated:♦←i X = �P

i X := {y ∈ W | ∀x ∈ W .(xRiy ⇒ x ∈ X )}. In otherwords, residuals are duals of conjugate operatorsSometimes, we can choose to add residuals/conjugatesexplicitly to the signature. That is, for every operator ♦i add anew operator ♦P

i satisfying

x ≤ �i♦Pi x

x ≤ �Pi ♦ix

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras

A BAO is a Lob algebra (or a diagonalizable algebra) if itsatisfies

♦x ≤ ♦(x ∧ ¬♦x)

� in Lob algebras corresponds to provability predicate Prov inPeano Arithmetics. And the name comes from

Corollary (of The Lob Theorem)

Prov(dProv(dψe) → ψe) → Prov(dψe) is provable in PeanoArithmetics for every formula ψ

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras

A BAO is a Lob algebra (or a diagonalizable algebra) if itsatisfies

♦x ≤ ♦(x ∧ ¬♦x)

� in Lob algebras corresponds to provability predicate Prov inPeano Arithmetics. And the name comes from

Corollary (of The Lob Theorem)

Prov(dProv(dψe) → ψe) → Prov(dψe) is provable in PeanoArithmetics for every formula ψ

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras and well-foundedness

Theorem

The complex BAO of 〈W ,R〉 is diagonalizable iff the converseof R is a strict well-founded order

An useful fact about Lob algebras is

Fact

For every x 6= ⊥ in a Lob algebra, x 6≤ ♦x

which is an algebraic reformulation of well-foundedness

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras and well-foundedness

Theorem

The complex BAO of 〈W ,R〉 is diagonalizable iff the converseof R is a strict well-founded order

An useful fact about Lob algebras is

Fact

For every x 6= ⊥ in a Lob algebra, x 6≤ ♦x

which is an algebraic reformulation of well-foundedness

Introduction Varieties not closed under completions Overview of completions Conclusions

Varieties

Let us recall fundamental operations on classes of algebras:H(X ) — the class of homomorphic images of algebras from X

S(X ) — the class of subalgebras of algebras from XP(X ) — the class of products of algebras from X

Theorem (Birkhoff-Tarski)

A class of algebras is a variety — i.e., is equationally definable— iff it is of the form HSP(X ) for some X

All classes of algebras introduced above — lattices, residuatedlattices, `-groups, FLew -algebras, MV-algebras, Heytingalgebras, Boolean algebras, BAOs, Lob algebras — areequationally definable

Introduction Varieties not closed under completions Overview of completions Conclusions

Varieties

Let us recall fundamental operations on classes of algebras:H(X ) — the class of homomorphic images of algebras from XS(X ) — the class of subalgebras of algebras from X

P(X ) — the class of products of algebras from X

Theorem (Birkhoff-Tarski)

A class of algebras is a variety — i.e., is equationally definable— iff it is of the form HSP(X ) for some X

All classes of algebras introduced above — lattices, residuatedlattices, `-groups, FLew -algebras, MV-algebras, Heytingalgebras, Boolean algebras, BAOs, Lob algebras — areequationally definable

Introduction Varieties not closed under completions Overview of completions Conclusions

Varieties

Let us recall fundamental operations on classes of algebras:H(X ) — the class of homomorphic images of algebras from XS(X ) — the class of subalgebras of algebras from XP(X ) — the class of products of algebras from X

Theorem (Birkhoff-Tarski)

A class of algebras is a variety — i.e., is equationally definable— iff it is of the form HSP(X ) for some X

All classes of algebras introduced above — lattices, residuatedlattices, `-groups, FLew -algebras, MV-algebras, Heytingalgebras, Boolean algebras, BAOs, Lob algebras — areequationally definable

Introduction Varieties not closed under completions Overview of completions Conclusions

Varieties

Let us recall fundamental operations on classes of algebras:H(X ) — the class of homomorphic images of algebras from XS(X ) — the class of subalgebras of algebras from XP(X ) — the class of products of algebras from X

Theorem (Birkhoff-Tarski)

A class of algebras is a variety — i.e., is equationally definable— iff it is of the form HSP(X ) for some X

All classes of algebras introduced above — lattices, residuatedlattices, `-groups, FLew -algebras, MV-algebras, Heytingalgebras, Boolean algebras, BAOs, Lob algebras — areequationally definable

Introduction Varieties not closed under completions Overview of completions Conclusions

Generation by complete algebras

Definition

A variety V of algebras with lattice reducts is completelygenerated if there is a class X of complete algebras s.t.V = HSP(X )

A very general notion, broader than, e.g., finite model property.Almost all classes of algebras introduced until now arecompletely generated . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

Generation by complete algebras

Definition

A variety V of algebras with lattice reducts is completelygenerated if there is a class X of complete algebras s.t.V = HSP(X )

A very general notion, broader than, e.g., finite model property.Almost all classes of algebras introduced until now arecompletely generated . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

. . . but not `-groups! Quite to the contrary: `-groups areanti-complete: this variety does not contain any non-trivialcomplete lattice

Fact (well-known?)

Every non-trivial `-group is unbounded

Proof.

Every non-trivial `-group contains y > 1. But then y2 > y

Introduction Varieties not closed under completions Overview of completions Conclusions

. . . but not `-groups! Quite to the contrary: `-groups areanti-complete: this variety does not contain any non-trivialcomplete lattice

Fact (well-known?)

Every non-trivial `-group is unbounded

Proof.

Every non-trivial `-group contains y > 1. But then y2 > y

Introduction Varieties not closed under completions Overview of completions Conclusions

. . . but not `-groups! Quite to the contrary: `-groups areanti-complete: this variety does not contain any non-trivialcomplete lattice

Fact (well-known?)

Every non-trivial `-group is unbounded

Proof.

Every non-trivial `-group contains y > 1. But then y2 > y

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras have finite model property and hence arecompletely generated

Theorem

Every Lob algebra can be obtained as a HSP-image of dualalgebras of finite strict orders

But when we enrich signature with ♦P , it is possible to obtain ananti-complete variety with diagonalizable reducts

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras have finite model property and hence arecompletely generated

Theorem

Every Lob algebra can be obtained as a HSP-image of dualalgebras of finite strict orders

But when we enrich signature with ♦P , it is possible to obtain ananti-complete variety with diagonalizable reducts

Introduction Varieties not closed under completions Overview of completions Conclusions

Lob algebras have finite model property and hence arecompletely generated

Theorem

Every Lob algebra can be obtained as a HSP-image of dualalgebras of finite strict orders

But when we enrich signature with ♦P , it is possible to obtain ananti-complete variety with diagonalizable reducts

Introduction Varieties not closed under completions Overview of completions Conclusions

Closure under completions

A variety V is closed under completions if there is a class ofcomplete algebras X s.t. V = IS(X )

It is much stronger notion than being completely generated. Itmeans that in generating the variety from its complete algebras,we can omit both H and P. As we will see soon, even fmp doesnot guarantee being completely generated. The converseimplication does not hold either, by the way. This notiongeneralizes the notion of canonicity — but this will be discussedfurther on

Introduction Varieties not closed under completions Overview of completions Conclusions

Closure under completions

A variety V is closed under completions if there is a class ofcomplete algebras X s.t. V = IS(X )It is much stronger notion than being completely generated. Itmeans that in generating the variety from its complete algebras,we can omit both H and P. As we will see soon, even fmp doesnot guarantee being completely generated. The converseimplication does not hold either, by the way.

This notiongeneralizes the notion of canonicity — but this will be discussedfurther on

Introduction Varieties not closed under completions Overview of completions Conclusions

Closure under completions

A variety V is closed under completions if there is a class ofcomplete algebras X s.t. V = IS(X )It is much stronger notion than being completely generated. Itmeans that in generating the variety from its complete algebras,we can omit both H and P. As we will see soon, even fmp doesnot guarantee being completely generated. The converseimplication does not hold either, by the way. This notiongeneralizes the notion of canonicity — but this will be discussedfurther on

Introduction Varieties not closed under completions Overview of completions Conclusions

Theorem

Lob algebras are not closed under completions

Theorem (with Tomasz Kowalski)

MV-algebras are not closed under completions

Introduction Varieties not closed under completions Overview of completions Conclusions

Theorem

Lob algebras are not closed under completions

Theorem (with Tomasz Kowalski)

MV-algebras are not closed under completions

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras I

Proof

Take 〈ω + ω∗, <〉: ω with the strict order followed by a copy of ωwith reverse order {0,1,2 . . . ,2∗,1∗,0∗}. This relationalstructure consists of the initial and final section of theultraproduct of all finite strict well-orders (ordinals)

Its dual algebra A is not a diagonalizable algebra. To see it,observe simply that ♦ω = ωNevertheless, the subalgebra B of A consisting of finite andcofinite subsets of A is a diagonalizable algebra

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras I

Proof

Take 〈ω + ω∗, <〉: ω with the strict order followed by a copy of ωwith reverse order {0,1,2 . . . ,2∗,1∗,0∗}. This relationalstructure consists of the initial and final section of theultraproduct of all finite strict well-orders (ordinals)Its dual algebra A is not a diagonalizable algebra. To see it,observe simply that ♦ω = ω

Nevertheless, the subalgebra B of A consisting of finite andcofinite subsets of A is a diagonalizable algebra

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras I

Proof

Take 〈ω + ω∗, <〉: ω with the strict order followed by a copy of ωwith reverse order {0,1,2 . . . ,2∗,1∗,0∗}. This relationalstructure consists of the initial and final section of theultraproduct of all finite strict well-orders (ordinals)Its dual algebra A is not a diagonalizable algebra. To see it,observe simply that ♦ω = ωNevertheless, the subalgebra B of A consisting of finite andcofinite subsets of A is a diagonalizable algebra

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras II

Proof (continued)

Assume now f : B 7→ C is an embedding in a completediagonalizable algebra. Denote cn := f ({n})

Then for every n ∈ ω, cn ≤ ♦cn+1. Hence

C :=∨n∈ω

cn ≤ ♦∨n∈ω

cn+1 = ♦C

(we don’t use complete additivity here!). C is not adiagonalizable algebra

Observe that B cannot be even embedded into any ω-completediagonalizable algebra. The proof works for any subvariety ofdiagonalizable algebras containing B

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras II

Proof (continued)

Assume now f : B 7→ C is an embedding in a completediagonalizable algebra. Denote cn := f ({n})Then for every n ∈ ω, cn ≤ ♦cn+1. Hence

C :=∨n∈ω

cn ≤ ♦∨n∈ω

cn+1 = ♦C

(we don’t use complete additivity here!). C is not adiagonalizable algebra

Observe that B cannot be even embedded into any ω-completediagonalizable algebra. The proof works for any subvariety ofdiagonalizable algebras containing B

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras II

Proof (continued)

Assume now f : B 7→ C is an embedding in a completediagonalizable algebra. Denote cn := f ({n})Then for every n ∈ ω, cn ≤ ♦cn+1. Hence

C :=∨n∈ω

cn ≤ ♦∨n∈ω

cn+1 = ♦C

(we don’t use complete additivity here!). C is not adiagonalizable algebra

Observe that B cannot be even embedded into any ω-completediagonalizable algebra. The proof works for any subvariety ofdiagonalizable algebras containing B

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for Lob algebras II

Proof (continued)

Assume now f : B 7→ C is an embedding in a completediagonalizable algebra. Denote cn := f ({n})Then for every n ∈ ω, cn ≤ ♦cn+1. Hence

C :=∨n∈ω

cn ≤ ♦∨n∈ω

cn+1 = ♦C

(we don’t use complete additivity here!). C is not adiagonalizable algebra

Observe that B cannot be even embedded into any ω-completediagonalizable algebra. The proof works for any subvariety ofdiagonalizable algebras containing B

Introduction Varieties not closed under completions Overview of completions Conclusions

Infinite distribtivity of MV-algebras

Lemma (with Nick Galatos)

In MV-algebras, x ·∧

yi =∧

x · yi

Sketch of proof

We don’t know how to show it directly for MV -algebras. We hadto use the fact that the lattice part of every MV-algebra iscompletely embeddable in a lattice of an `-group. Thisembedding does not preserve ·. It satisfies instead

f (x · y) = f (x) · f (y) ∨ f (0)

— multiplication of MV-algebras is a truncated variant of`-group multiplication. Using the two infinite distibutive laws of`-groups mentioned before, the result follows

Introduction Varieties not closed under completions Overview of completions Conclusions

Infinite distribtivity of MV-algebras

Lemma (with Nick Galatos)

In MV-algebras, x ·∧

yi =∧

x · yi

Sketch of proof

We don’t know how to show it directly for MV -algebras. We hadto use the fact that the lattice part of every MV-algebra iscompletely embeddable in a lattice of an `-group.

Thisembedding does not preserve ·. It satisfies instead

f (x · y) = f (x) · f (y) ∨ f (0)

— multiplication of MV-algebras is a truncated variant of`-group multiplication. Using the two infinite distibutive laws of`-groups mentioned before, the result follows

Introduction Varieties not closed under completions Overview of completions Conclusions

Infinite distribtivity of MV-algebras

Lemma (with Nick Galatos)

In MV-algebras, x ·∧

yi =∧

x · yi

Sketch of proof

We don’t know how to show it directly for MV -algebras. We hadto use the fact that the lattice part of every MV-algebra iscompletely embeddable in a lattice of an `-group. Thisembedding does not preserve ·. It satisfies instead

f (x · y) = f (x) · f (y) ∨ f (0)

— multiplication of MV-algebras is a truncated variant of`-group multiplication.

Using the two infinite distibutive laws of`-groups mentioned before, the result follows

Introduction Varieties not closed under completions Overview of completions Conclusions

Infinite distribtivity of MV-algebras

Lemma (with Nick Galatos)

In MV-algebras, x ·∧

yi =∧

x · yi

Sketch of proof

We don’t know how to show it directly for MV -algebras. We hadto use the fact that the lattice part of every MV-algebra iscompletely embeddable in a lattice of an `-group. Thisembedding does not preserve ·. It satisfies instead

f (x · y) = f (x) · f (y) ∨ f (0)

— multiplication of MV-algebras is a truncated variant of`-group multiplication. Using the two infinite distibutive laws of`-groups mentioned before, the result follows

Introduction Varieties not closed under completions Overview of completions Conclusions

MV-algebras are not closed under completions

Now define residuation on ω + ω∗ as follows:

n ·m := ⊥ n∗ ·m := m ·− n n∗ ·m∗ := (n + m)∗

where ·− stands for truncated substraction. Let C be theMV-algebra obtained this way; C is known as Chang’s Chain

Theorem

C cannot be embedded into any complete MV-algebra

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for MV algebras

Proof.

Assume f : C 7→ D is such an embedding. Let an := f (n∗),bn := f (n), a∞ :=

∧n∈ω

an, b∞ :=∨

n∈ωbn

By infinite distributivity for∨

, a∞ · b∞ =∨

n∈ω(a∞ · bn). For every

nω, a∞ ≤ an = bn → ⊥. Hence, a∞ · b∞ = ⊥But by infinite distributivity for

∧, b∞ · a∞ =

∧n∈ω

(b∞ · an). For

every n ∈ ω, b∞ · an =∨

m∈ω(bm · an) =

∨m∈ω

bm ·−n = b∞, a

contradiction.

This proof works for any subvariety of FLew which contains C

and satisfies infinite distributivity law for∧

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for MV algebras

Proof.

Assume f : C 7→ D is such an embedding. Let an := f (n∗),bn := f (n), a∞ :=

∧n∈ω

an, b∞ :=∨

n∈ωbn

By infinite distributivity for∨

, a∞ · b∞ =∨

n∈ω(a∞ · bn). For every

nω, a∞ ≤ an = bn → ⊥. Hence, a∞ · b∞ = ⊥

But by infinite distributivity for∧

, b∞ · a∞ =∧

n∈ω(b∞ · an). For

every n ∈ ω, b∞ · an =∨

m∈ω(bm · an) =

∨m∈ω

bm ·−n = b∞, a

contradiction.

This proof works for any subvariety of FLew which contains C

and satisfies infinite distributivity law for∧

Introduction Varieties not closed under completions Overview of completions Conclusions

Proof for MV algebras

Proof.

Assume f : C 7→ D is such an embedding. Let an := f (n∗),bn := f (n), a∞ :=

∧n∈ω

an, b∞ :=∨

n∈ωbn

By infinite distributivity for∨

, a∞ · b∞ =∨

n∈ω(a∞ · bn). For every

nω, a∞ ≤ an = bn → ⊥. Hence, a∞ · b∞ = ⊥But by infinite distributivity for

∧, b∞ · a∞ =

∧n∈ω

(b∞ · an). For

every n ∈ ω, b∞ · an =∨

m∈ω(bm · an) =

∨m∈ω

bm ·−n = b∞, a

contradiction.

This proof works for any subvariety of FLew which contains C

and satisfies infinite distributivity law for∧

Introduction Varieties not closed under completions Overview of completions Conclusions

Comments

These two proofs are surprisingly similar. The main differenceis perhaps that in case of proof for Lob algebras we have notused infinite distributivity laws.

Both examples were constructedby taking the initial and final part of an ultraproduct of finitesubdirectly irreducible algebras and by showing that supremumof the tail part and/or infimum of the head part must misbehaveDoes it have any connection with the fact that both Lob algebrasand MV-algebras are closed under relativization? Given anyelement x , take [x) to be the new universe and define

�xa := (�a ∨ x) or a ·x b := a · b ∨ x

♦ (or →) modified accordingly. You get a Lob algebra (orMV-algebra) again. Perhaps such an elastic structure alwayshas to show some gaps when stretched? Maybe topology ornon-standard analysis can explain what’s going on?

Introduction Varieties not closed under completions Overview of completions Conclusions

Comments

These two proofs are surprisingly similar. The main differenceis perhaps that in case of proof for Lob algebras we have notused infinite distributivity laws. Both examples were constructedby taking the initial and final part of an ultraproduct of finitesubdirectly irreducible algebras and by showing that supremumof the tail part and/or infimum of the head part must misbehave

Does it have any connection with the fact that both Lob algebrasand MV-algebras are closed under relativization? Given anyelement x , take [x) to be the new universe and define

�xa := (�a ∨ x) or a ·x b := a · b ∨ x

♦ (or →) modified accordingly. You get a Lob algebra (orMV-algebra) again. Perhaps such an elastic structure alwayshas to show some gaps when stretched? Maybe topology ornon-standard analysis can explain what’s going on?

Introduction Varieties not closed under completions Overview of completions Conclusions

Comments

These two proofs are surprisingly similar. The main differenceis perhaps that in case of proof for Lob algebras we have notused infinite distributivity laws. Both examples were constructedby taking the initial and final part of an ultraproduct of finitesubdirectly irreducible algebras and by showing that supremumof the tail part and/or infimum of the head part must misbehaveDoes it have any connection with the fact that both Lob algebrasand MV-algebras are closed under relativization?

Given anyelement x , take [x) to be the new universe and define

�xa := (�a ∨ x) or a ·x b := a · b ∨ x

♦ (or →) modified accordingly. You get a Lob algebra (orMV-algebra) again. Perhaps such an elastic structure alwayshas to show some gaps when stretched? Maybe topology ornon-standard analysis can explain what’s going on?

Introduction Varieties not closed under completions Overview of completions Conclusions

Comments

These two proofs are surprisingly similar. The main differenceis perhaps that in case of proof for Lob algebras we have notused infinite distributivity laws. Both examples were constructedby taking the initial and final part of an ultraproduct of finitesubdirectly irreducible algebras and by showing that supremumof the tail part and/or infimum of the head part must misbehaveDoes it have any connection with the fact that both Lob algebrasand MV-algebras are closed under relativization? Given anyelement x , take [x) to be the new universe and define

�xa := (�a ∨ x) or a ·x b := a · b ∨ x

♦ (or →) modified accordingly.

You get a Lob algebra (orMV-algebra) again. Perhaps such an elastic structure alwayshas to show some gaps when stretched? Maybe topology ornon-standard analysis can explain what’s going on?

Introduction Varieties not closed under completions Overview of completions Conclusions

Comments

These two proofs are surprisingly similar. The main differenceis perhaps that in case of proof for Lob algebras we have notused infinite distributivity laws. Both examples were constructedby taking the initial and final part of an ultraproduct of finitesubdirectly irreducible algebras and by showing that supremumof the tail part and/or infimum of the head part must misbehaveDoes it have any connection with the fact that both Lob algebrasand MV-algebras are closed under relativization? Given anyelement x , take [x) to be the new universe and define

�xa := (�a ∨ x) or a ·x b := a · b ∨ x

♦ (or →) modified accordingly. You get a Lob algebra (orMV-algebra) again. Perhaps such an elastic structure alwayshas to show some gaps when stretched? Maybe topology ornon-standard analysis can explain what’s going on?

Introduction Varieties not closed under completions Overview of completions Conclusions

At any rate, these proofs are based on rather simplecalculations. It is surprising to learn they generalize twoorignally more involved results: that neither diagonalizablealgebras nor MV algebras are canonical, the latter resultobtained by Gehrke and Priestley. In order to understand betterthe notion of closure under completions, let us review nowsome completion constructions from the literature

Introduction Varieties not closed under completions Overview of completions Conclusions

Dedekind-MacNeille completion

Definition

For a subset P of a poset, let L(P) be the set of lower boundsfor P, U(P) be the subset of upper bounds of P. A set A is anormal ideal if A = LU(A)

Fact

In a lattice, every normal ideal is also an ideal in normal sense.Every principal ideal is normal

Definition

An ideal is complete if it is closed under existing joins

Fact

Every normal ideal is complete. In Heyting algebras, theconverse also holds

Introduction Varieties not closed under completions Overview of completions Conclusions

Dedekind-MacNeille completion

Definition

For a subset P of a poset, let L(P) be the set of lower boundsfor P, U(P) be the subset of upper bounds of P. A set A is anormal ideal if A = LU(A)

Fact

In a lattice, every normal ideal is also an ideal in normal sense.Every principal ideal is normal

Definition

An ideal is complete if it is closed under existing joins

Fact

Every normal ideal is complete. In Heyting algebras, theconverse also holds

Introduction Varieties not closed under completions Overview of completions Conclusions

Dedekind-MacNeille completion

Definition

For a subset P of a poset, let L(P) be the set of lower boundsfor P, U(P) be the subset of upper bounds of P. A set A is anormal ideal if A = LU(A)

Fact

In a lattice, every normal ideal is also an ideal in normal sense.Every principal ideal is normal

Definition

An ideal is complete if it is closed under existing joins

Fact

Every normal ideal is complete. In Heyting algebras, theconverse also holds

Introduction Varieties not closed under completions Overview of completions Conclusions

Dedekind-MacNeille completion

Definition

For a subset P of a poset, let L(P) be the set of lower boundsfor P, U(P) be the subset of upper bounds of P. A set A is anormal ideal if A = LU(A)

Fact

In a lattice, every normal ideal is also an ideal in normal sense.Every principal ideal is normal

Definition

An ideal is complete if it is closed under existing joins

Fact

Every normal ideal is complete. In Heyting algebras, theconverse also holds

Introduction Varieties not closed under completions Overview of completions Conclusions

Theorem

For every poset P, the collection of all normal ideals of P formsa complete lattice MacN(P) with

∧N =

⋂N and∨

N = LU(⋃

N). P is embeddable in MacN(P) by f (p) = (p].MacN(P) is called the Dedekind-MacNeille completion of P. fpreserves all existing joins and meets

As we shall see, Heyting algebras and Boolean algebras areMacN-closed, the latter fact known already by MacNeillehimself. Distributive lattices are not. This has been proven byFunayama in the 1940’s

Introduction Varieties not closed under completions Overview of completions Conclusions

Theorem

For every poset P, the collection of all normal ideals of P formsa complete lattice MacN(P) with

∧N =

⋂N and∨

N = LU(⋃

N). P is embeddable in MacN(P) by f (p) = (p].MacN(P) is called the Dedekind-MacNeille completion of P. fpreserves all existing joins and meets

As we shall see, Heyting algebras and Boolean algebras areMacN-closed, the latter fact known already by MacNeillehimself. Distributive lattices are not. This has been proven byFunayama in the 1940’s

Introduction Varieties not closed under completions Overview of completions Conclusions

A more abstract characterization

It is known since Banaschewski and Schmidt thatDedekind-MacNeille completions have the following, moreabstract characterization

Definition

Let P be any subset of a complete lattice L. We say that P ismeet-dense in L if every element of L can be obtained as meetof elements of P. The notion of join-density is defined dually

Theorem

For every poset P there is, up to isomorphism, exactly onecomplete lattice L s.t. P is both meet-dense and join-dense inL. This lattice is MacN(P)

Introduction Varieties not closed under completions Overview of completions Conclusions

A more abstract characterization

It is known since Banaschewski and Schmidt thatDedekind-MacNeille completions have the following, moreabstract characterization

Definition

Let P be any subset of a complete lattice L. We say that P ismeet-dense in L if every element of L can be obtained as meetof elements of P. The notion of join-density is defined dually

Theorem

For every poset P there is, up to isomorphism, exactly onecomplete lattice L s.t. P is both meet-dense and join-dense inL. This lattice is MacN(P)

Introduction Varieties not closed under completions Overview of completions Conclusions

MacNeille completions of Heyting algebras

Assume L is a Heyting alebra. As the lattice structure ofMacN(L) is completely determined by L, we have no realchoice how to define implication for every P = LU(P),R = LU(P):

P → R := {s ∈ L | ∀p ∈ P,p ∧ s ∈ R}

This gives you a Heyting algebra: P → R is well-defined and isthe pseudocomplement of P with respect to R. So Heytingalgebras are closed under MacNeille completions, but . . .

Theorem (Harding and Bezhanishvili)

The only subvariety of Heyting algebras which is MacN-closedis the variety of boolean algebras. In other words, MacNeillecompletions do not preserve any non-trivial Heyting equations

Introduction Varieties not closed under completions Overview of completions Conclusions

MacNeille completions of Heyting algebras

Assume L is a Heyting alebra. As the lattice structure ofMacN(L) is completely determined by L, we have no realchoice how to define implication for every P = LU(P),R = LU(P):

P → R := {s ∈ L | ∀p ∈ P,p ∧ s ∈ R}

This gives you a Heyting algebra: P → R is well-defined and isthe pseudocomplement of P with respect to R. So Heytingalgebras are closed under MacNeille completions, but . . .

Theorem (Harding and Bezhanishvili)

The only subvariety of Heyting algebras which is MacN-closedis the variety of boolean algebras. In other words, MacNeillecompletions do not preserve any non-trivial Heyting equations

Introduction Varieties not closed under completions Overview of completions Conclusions

MacNeille completions of Heyting algebras

Assume L is a Heyting alebra. As the lattice structure ofMacN(L) is completely determined by L, we have no realchoice how to define implication for every P = LU(P),R = LU(P):

P → R := {s ∈ L | ∀p ∈ P,p ∧ s ∈ R}

This gives you a Heyting algebra: P → R is well-defined and isthe pseudocomplement of P with respect to R. So Heytingalgebras are closed under MacNeille completions, but . . .

Theorem (Harding and Bezhanishvili)

The only subvariety of Heyting algebras which is MacN-closedis the variety of boolean algebras. In other words, MacNeillecompletions do not preserve any non-trivial Heyting equations

Introduction Varieties not closed under completions Overview of completions Conclusions

Now, what if L contains some additional operators, e.g., L is aBAO?

We have to choose between two definitions

Definition

Let a BAO A be a dense subalgebra of a complete booleanalgebra B (i.e., B ' A). Define the lower MacNeile completion

♦↓b :=∨{♦a | a ≤ b})

and the upper MacNeile completion

♦↑b :=∧{♦a | b ≤ a})

Similar definitions can be formulated for arbitraryorder-preserving or order-reserving operations on lattices: cf.an exhaustive recent study of Theunissen and Venema. Thequestion is: Which of these definitions gives you an operatoragain — that is, which one distributes over finite joins?

Introduction Varieties not closed under completions Overview of completions Conclusions

Now, what if L contains some additional operators, e.g., L is aBAO? We have to choose between two definitions

Definition

Let a BAO A be a dense subalgebra of a complete booleanalgebra B (i.e., B ' A). Define the lower MacNeile completion

♦↓b :=∨{♦a | a ≤ b})

and the upper MacNeile completion

♦↑b :=∧{♦a | b ≤ a})

Similar definitions can be formulated for arbitraryorder-preserving or order-reserving operations on lattices: cf.an exhaustive recent study of Theunissen and Venema. Thequestion is: Which of these definitions gives you an operatoragain — that is, which one distributes over finite joins?

Introduction Varieties not closed under completions Overview of completions Conclusions

Now, what if L contains some additional operators, e.g., L is aBAO? We have to choose between two definitions

Definition

Let a BAO A be a dense subalgebra of a complete booleanalgebra B (i.e., B ' A). Define the lower MacNeile completion

♦↓b :=∨{♦a | a ≤ b})

and the upper MacNeile completion

♦↑b :=∧{♦a | b ≤ a})

Similar definitions can be formulated for arbitraryorder-preserving or order-reserving operations on lattices: cf.an exhaustive recent study of Theunissen and Venema. Thequestion is: Which of these definitions gives you an operatoragain — that is, which one distributes over finite joins?

Introduction Varieties not closed under completions Overview of completions Conclusions

Now, what if L contains some additional operators, e.g., L is aBAO? We have to choose between two definitions

Definition

Let a BAO A be a dense subalgebra of a complete booleanalgebra B (i.e., B ' A). Define the lower MacNeile completion

♦↓b :=∨{♦a | a ≤ b})

and the upper MacNeile completion

♦↑b :=∧{♦a | b ≤ a})

Similar definitions can be formulated for arbitraryorder-preserving or order-reserving operations on lattices: cf.an exhaustive recent study of Theunissen and Venema. Thequestion is: Which of these definitions gives you an operatoragain — that is, which one distributes over finite joins?

Introduction Varieties not closed under completions Overview of completions Conclusions

Now, what if L contains some additional operators, e.g., L is aBAO? We have to choose between two definitions

Definition

Let a BAO A be a dense subalgebra of a complete booleanalgebra B (i.e., B ' A). Define the lower MacNeile completion

♦↓b :=∨{♦a | a ≤ b})

and the upper MacNeile completion

♦↑b :=∧{♦a | b ≤ a})

Similar definitions can be formulated for arbitraryorder-preserving or order-reserving operations on lattices: cf.an exhaustive recent study of Theunissen and Venema. Thequestion is: Which of these definitions gives you an operatoragain — that is, which one distributes over finite joins?

Introduction Varieties not closed under completions Overview of completions Conclusions

To see that ♦↓ can be nasty, consider the boolean algebra offinite and cofinite sets over ω and ♦↓ for ♦ mapping cofinite setsto > and finite to ⊥.

But we have the following

Fact

For every operator ♦, ♦↑ is again an operator

Introduction Varieties not closed under completions Overview of completions Conclusions

To see that ♦↓ can be nasty, consider the boolean algebra offinite and cofinite sets over ω and ♦↓ for ♦ mapping cofinite setsto > and finite to ⊥. But we have the following

Fact

For every operator ♦, ♦↑ is again an operator

Introduction Varieties not closed under completions Overview of completions Conclusions

Why MacNeile completions are useful?

Their most important feature is perhaps preservation of existing∧’s and

∨’s. Why we may desire something like this?

Completeness proofs for predicate and infinitary logic

Recall the idea of boolean-valued models: set-theorists in theaudience know it far better than myself. We use infima andsuprema to interpret quantifiers

Introduction Varieties not closed under completions Overview of completions Conclusions

Why MacNeile completions are useful?

Their most important feature is perhaps preservation of existing∧’s and

∨’s. Why we may desire something like this?

Completeness proofs for predicate and infinitary logic

Recall the idea of boolean-valued models: set-theorists in theaudience know it far better than myself. We use infima andsuprema to interpret quantifiers

Introduction Varieties not closed under completions Overview of completions Conclusions

Predicate completeness via MacN-closure

Roughly speaking, the idea is as follows: assume thepropositional fragment of a logic L corresponds to variety V :that is, AL — the Lindenbaum-Tarski algebra of L —HSP-generates V . Assume also V is closed under MacN.

Quantifiers correspond to existing suprema and infima in AL

and hence they are preserved in MacNAL. Using this fact, onecan obtain a proof of completeness of L with respect toMacNAL-valued models. This idea for classical logic originatedalready in Rasiowa and Sikorski; see papers of Ono foroverview in non-classical logics. Another related example (cf.Pratt): ♦ completely additive and ♦∗ satisfying

♦∗x :=∨n∈ω

♦nx

In general, we need MacN — not just arbitrary completions —to preserve this property

Introduction Varieties not closed under completions Overview of completions Conclusions

Predicate completeness via MacN-closure

Roughly speaking, the idea is as follows: assume thepropositional fragment of a logic L corresponds to variety V :that is, AL — the Lindenbaum-Tarski algebra of L —HSP-generates V . Assume also V is closed under MacN.Quantifiers correspond to existing suprema and infima in AL

and hence they are preserved in MacNAL.

Using this fact, onecan obtain a proof of completeness of L with respect toMacNAL-valued models. This idea for classical logic originatedalready in Rasiowa and Sikorski; see papers of Ono foroverview in non-classical logics. Another related example (cf.Pratt): ♦ completely additive and ♦∗ satisfying

♦∗x :=∨n∈ω

♦nx

In general, we need MacN — not just arbitrary completions —to preserve this property

Introduction Varieties not closed under completions Overview of completions Conclusions

Predicate completeness via MacN-closure

Roughly speaking, the idea is as follows: assume thepropositional fragment of a logic L corresponds to variety V :that is, AL — the Lindenbaum-Tarski algebra of L —HSP-generates V . Assume also V is closed under MacN.Quantifiers correspond to existing suprema and infima in AL

and hence they are preserved in MacNAL. Using this fact, onecan obtain a proof of completeness of L with respect toMacNAL-valued models. This idea for classical logic originatedalready in Rasiowa and Sikorski; see papers of Ono foroverview in non-classical logics.

Another related example (cf.Pratt): ♦ completely additive and ♦∗ satisfying

♦∗x :=∨n∈ω

♦nx

In general, we need MacN — not just arbitrary completions —to preserve this property

Introduction Varieties not closed under completions Overview of completions Conclusions

Predicate completeness via MacN-closure

Roughly speaking, the idea is as follows: assume thepropositional fragment of a logic L corresponds to variety V :that is, AL — the Lindenbaum-Tarski algebra of L —HSP-generates V . Assume also V is closed under MacN.Quantifiers correspond to existing suprema and infima in AL

and hence they are preserved in MacNAL. Using this fact, onecan obtain a proof of completeness of L with respect toMacNAL-valued models. This idea for classical logic originatedalready in Rasiowa and Sikorski; see papers of Ono foroverview in non-classical logics. Another related example (cf.Pratt): ♦ completely additive and ♦∗ satisfying

♦∗x :=∨n∈ω

♦nx

In general, we need MacN — not just arbitrary completions —to preserve this property

Introduction Varieties not closed under completions Overview of completions Conclusions

Predicate completeness via MacN-closure

Roughly speaking, the idea is as follows: assume thepropositional fragment of a logic L corresponds to variety V :that is, AL — the Lindenbaum-Tarski algebra of L —HSP-generates V . Assume also V is closed under MacN.Quantifiers correspond to existing suprema and infima in AL

and hence they are preserved in MacNAL. Using this fact, onecan obtain a proof of completeness of L with respect toMacNAL-valued models. This idea for classical logic originatedalready in Rasiowa and Sikorski; see papers of Ono foroverview in non-classical logics. Another related example (cf.Pratt): ♦ completely additive and ♦∗ satisfying

♦∗x :=∨n∈ω

♦nx

In general, we need MacN — not just arbitrary completions —to preserve this property

Introduction Varieties not closed under completions Overview of completions Conclusions

Power and weakness of MacNeille completions

What are the problems, then?

1 As was mentioned earlier, it is very hard for a variety to beclosed under MacNeille completions — subvarieties ofHeyting algebras being the most extreme example. Thesituation with modal algebras is not much better either,although Givant and Venema were able to prove thatcertain very well-behaved equalities can be preserved

2 Preservation of∧

and∨

is sometimes an advantage,sometimes a problem. As you recall, in the BAO case thedual algebras of Kripke frames were not only complete, butalso atomic. But Dedekind-MacNeille closure of anyalgebra has only as many atoms as the original algebraitself: it does not add any “separating points”. If we needconcrete, set-theoretical representations, we have to workwith a different kind of completions.

Introduction Varieties not closed under completions Overview of completions Conclusions

Power and weakness of MacNeille completions

What are the problems, then?1 As was mentioned earlier, it is very hard for a variety to be

closed under MacNeille completions — subvarieties ofHeyting algebras being the most extreme example. Thesituation with modal algebras is not much better either,although Givant and Venema were able to prove thatcertain very well-behaved equalities can be preserved

2 Preservation of∧

and∨

is sometimes an advantage,sometimes a problem. As you recall, in the BAO case thedual algebras of Kripke frames were not only complete, butalso atomic. But Dedekind-MacNeille closure of anyalgebra has only as many atoms as the original algebraitself: it does not add any “separating points”. If we needconcrete, set-theoretical representations, we have to workwith a different kind of completions.

Introduction Varieties not closed under completions Overview of completions Conclusions

Power and weakness of MacNeille completions

What are the problems, then?1 As was mentioned earlier, it is very hard for a variety to be

closed under MacNeille completions — subvarieties ofHeyting algebras being the most extreme example. Thesituation with modal algebras is not much better either,although Givant and Venema were able to prove thatcertain very well-behaved equalities can be preserved

2 Preservation of∧

and∨

is sometimes an advantage,sometimes a problem. As you recall, in the BAO case thedual algebras of Kripke frames were not only complete, butalso atomic. But Dedekind-MacNeille closure of anyalgebra has only as many atoms as the original algebraitself: it does not add any “separating points”. If we needconcrete, set-theoretical representations, we have to workwith a different kind of completions.

Introduction Varieties not closed under completions Overview of completions Conclusions

Canonical completions

While the Dedekind-MacNeille completion originated in thetheory of order and there is nothing specifically boolean to it,the next kind of completion — canonical completion — arisedfrom the theory of boolean algebras and was generalized onlymuch later to non-boolean setting.

Therefore, we begin bydiscussing this notion in the boolean context and only later weshow briefly — if time allows — how it can be generalized forother kinds of algebras.Recall first that every element of a boolean algebra can beidentified with the set of ultrafilters it belongs to. This is thecontents of the famous Stone representation theorem:ultrafilters can be thought of as non-standard points

Introduction Varieties not closed under completions Overview of completions Conclusions

Canonical completions

While the Dedekind-MacNeille completion originated in thetheory of order and there is nothing specifically boolean to it,the next kind of completion — canonical completion — arisedfrom the theory of boolean algebras and was generalized onlymuch later to non-boolean setting. Therefore, we begin bydiscussing this notion in the boolean context and only later weshow briefly — if time allows — how it can be generalized forother kinds of algebras.Recall first that every element of a boolean algebra can beidentified with the set of ultrafilters it belongs to. This is thecontents of the famous Stone representation theorem:ultrafilters can be thought of as non-standard points

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone space

More formally, let A be a boolean algebra, Uf (A) be the set ofall its ultrafilters and let π : A 3 a 7→ {∆ | a ∈ ∆} ⊆ Uf (A).

Define the base of open sets over Uf (A) to be {f (a) | a ∈ A}and the space thus obtained by T. Then

Theorem (Stone)

1 T has a basis of clopen sets, is compact and Hausdorff.Hence, it is totally disconnected

2 π is a boolean isomorphism of A onto the family of clopensets of T. What follows, π[A] is a boolean subalgebra ofAσ := ℘(Uf (A)) isomorphic to A itself

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone space

More formally, let A be a boolean algebra, Uf (A) be the set ofall its ultrafilters and let π : A 3 a 7→ {∆ | a ∈ ∆} ⊆ Uf (A).Define the base of open sets over Uf (A) to be {f (a) | a ∈ A}and the space thus obtained by T.

Then

Theorem (Stone)

1 T has a basis of clopen sets, is compact and Hausdorff.Hence, it is totally disconnected

2 π is a boolean isomorphism of A onto the family of clopensets of T. What follows, π[A] is a boolean subalgebra ofAσ := ℘(Uf (A)) isomorphic to A itself

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone space

More formally, let A be a boolean algebra, Uf (A) be the set ofall its ultrafilters and let π : A 3 a 7→ {∆ | a ∈ ∆} ⊆ Uf (A).Define the base of open sets over Uf (A) to be {f (a) | a ∈ A}and the space thus obtained by T. Then

Theorem (Stone)

1 T has a basis of clopen sets, is compact and Hausdorff.Hence, it is totally disconnected

2 π is a boolean isomorphism of A onto the family of clopensets of T. What follows, π[A] is a boolean subalgebra ofAσ := ℘(Uf (A)) isomorphic to A itself

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone space

More formally, let A be a boolean algebra, Uf (A) be the set ofall its ultrafilters and let π : A 3 a 7→ {∆ | a ∈ ∆} ⊆ Uf (A).Define the base of open sets over Uf (A) to be {f (a) | a ∈ A}and the space thus obtained by T. Then

Theorem (Stone)

1 T has a basis of clopen sets, is compact and Hausdorff.Hence, it is totally disconnected

2 π is a boolean isomorphism of A onto the family of clopensets of T. What follows, π[A] is a boolean subalgebra ofAσ := ℘(Uf (A)) isomorphic to A itself

Introduction Varieties not closed under completions Overview of completions Conclusions

Recall that we were able to characterize abstractlyDedekind-MacNeille completions as the only meet- andjoin-dense ones. Do we have an analogous characterization ofStone representation?

Definition

Let L be a complete lattice, P — a subset of its universe. P iscompact in L if for every A,B ⊆ P,

∧A ≤

∨B implies existence

of some A0 ⊆fin A and B0 ⊆fin B s.t.∧

A0 ≤∨

B0

Definition

P is dense in L iff for every a ∈ L there are families{Ai}i∈J , {Bi}i∈J ⊆ P s.t.

a =∧i∈I

∨Ai =

∨i∈I

∧Bi

Introduction Varieties not closed under completions Overview of completions Conclusions

Recall that we were able to characterize abstractlyDedekind-MacNeille completions as the only meet- andjoin-dense ones. Do we have an analogous characterization ofStone representation?

Definition

Let L be a complete lattice, P — a subset of its universe. P iscompact in L if for every A,B ⊆ P,

∧A ≤

∨B implies existence

of some A0 ⊆fin A and B0 ⊆fin B s.t.∧

A0 ≤∨

B0

Definition

P is dense in L iff for every a ∈ L there are families{Ai}i∈J , {Bi}i∈J ⊆ P s.t.

a =∧i∈I

∨Ai =

∨i∈I

∧Bi

Introduction Varieties not closed under completions Overview of completions Conclusions

Recall that we were able to characterize abstractlyDedekind-MacNeille completions as the only meet- andjoin-dense ones. Do we have an analogous characterization ofStone representation?

Definition

Let L be a complete lattice, P — a subset of its universe. P iscompact in L if for every A,B ⊆ P,

∧A ≤

∨B implies existence

of some A0 ⊆fin A and B0 ⊆fin B s.t.∧

A0 ≤∨

B0

Definition

P is dense in L iff for every a ∈ L there are families{Ai}i∈J , {Bi}i∈J ⊆ P s.t.

a =∧i∈I

∨Ai =

∨i∈I

∧Bi

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone representation is compact and dense

Theorem

π[A] is compact in Aσ

π[A] is dense in Aσ

Proof.

By the Prime Ideal Theorem

Every atom Aσ is a meet of elements from π[A]. Dually,every coatom is a join of elements from π[A]

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone representation is compact and dense

Theorem

π[A] is compact in Aσ

π[A] is dense in Aσ

Proof.

By the Prime Ideal Theorem

Every atom Aσ is a meet of elements from π[A]. Dually,every coatom is a join of elements from π[A]

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone representation is compact and dense

Theorem

π[A] is compact in Aσ

π[A] is dense in Aσ

Proof.

By the Prime Ideal Theorem

Every atom Aσ is a meet of elements from π[A]. Dually,every coatom is a join of elements from π[A]

Introduction Varieties not closed under completions Overview of completions Conclusions

Stone representation is compact and dense

Theorem

π[A] is compact in Aσ

π[A] is dense in Aσ

Proof.

By the Prime Ideal Theorem

Every atom Aσ is a meet of elements from π[A]. Dually,every coatom is a join of elements from π[A]

Introduction Varieties not closed under completions Overview of completions Conclusions

Abstract characterization

Yes, you guessed it:

Theorem

Every compact and dense completion of A is isomorphic to Aσ

Gehrke and Harding took this to be the starting point of theirabstract characterization of canonical extensions for arbitrarylattices

Introduction Varieties not closed under completions Overview of completions Conclusions

Abstract characterization

Yes, you guessed it:

Theorem

Every compact and dense completion of A is isomorphic to Aσ

Gehrke and Harding took this to be the starting point of theirabstract characterization of canonical extensions for arbitrarylattices

Introduction Varieties not closed under completions Overview of completions Conclusions

Abstract characterization

Yes, you guessed it:

Theorem

Every compact and dense completion of A is isomorphic to Aσ

Gehrke and Harding took this to be the starting point of theirabstract characterization of canonical extensions for arbitrarylattices

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:

∨ ∧: every set of ultrafilters is the sum of closed elements,

i.e., singletons of its own elements. For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}. Then for any

collection of ultrafilters X , define ♦σX :=⋃{♦σ{Γ} | Γ ∈ X}∧ ∨

: dually, taking intersection of operators defined oncoatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:∨ ∧

: every set of ultrafilters is the sum of closed elements,i.e., singletons of its own elements.

For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}. Then for any

collection of ultrafilters X , define ♦σX :=⋃{♦σ{Γ} | Γ ∈ X}∧ ∨

: dually, taking intersection of operators defined oncoatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:∨ ∧

: every set of ultrafilters is the sum of closed elements,i.e., singletons of its own elements. For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}.

Then for anycollection of ultrafilters X , define ♦σX :=

⋃{♦σ{Γ} | Γ ∈ X}∧ ∨

: dually, taking intersection of operators defined oncoatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:∨ ∧

: every set of ultrafilters is the sum of closed elements,i.e., singletons of its own elements. For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}. Then for any

collection of ultrafilters X , define ♦σX :=⋃{♦σ{Γ} | Γ ∈ X}

∧ ∨: dually, taking intersection of operators defined on

coatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:∨ ∧

: every set of ultrafilters is the sum of closed elements,i.e., singletons of its own elements. For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}. Then for any

collection of ultrafilters X , define ♦σX :=⋃{♦σ{Γ} | Γ ∈ X}∧ ∨

: dually, taking intersection of operators defined oncoatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

How to define operators on the Stone space?

Just like join-density and meet-density suggested the definitionof lower and upper MacNeille completion, density suggests thenotion of lower and upper canonical extension:∨ ∧

: every set of ultrafilters is the sum of closed elements,i.e., singletons of its own elements. For a single ultrafilterΓ, define ♦σ{Γ} :=

⋂{π(♦a) | a ∈ Γ}. Then for any

collection of ultrafilters X , define ♦σX :=⋃{♦σ{Γ} | Γ ∈ X}∧ ∨

: dually, taking intersection of operators defined oncoatoms

Before we discuss it further, though, let us pause for a whileand ask . . .

Introduction Varieties not closed under completions Overview of completions Conclusions

The Necessary Condition

. . . what exactly do we want to achieve?

Definition (Necessary Condition)

The Necessary Condition for a good operator f on Aσ: for everya ∈ A, f (π(a)) = π(♦a)

In other words, we simply want π to a BAO-embedding, not justa boolean algebra embedding. The operator ♦σ defined on theprevious slide satisfies The Necessary Condition. But it is by nomeans clear that this is the only operator which satisfies it!

Introduction Varieties not closed under completions Overview of completions Conclusions

The Necessary Condition

. . . what exactly do we want to achieve?

Definition (Necessary Condition)

The Necessary Condition for a good operator f on Aσ:

for everya ∈ A, f (π(a)) = π(♦a)

In other words, we simply want π to a BAO-embedding, not justa boolean algebra embedding. The operator ♦σ defined on theprevious slide satisfies The Necessary Condition. But it is by nomeans clear that this is the only operator which satisfies it!

Introduction Varieties not closed under completions Overview of completions Conclusions

The Necessary Condition

. . . what exactly do we want to achieve?

Definition (Necessary Condition)

The Necessary Condition for a good operator f on Aσ: for everya ∈ A, f (π(a)) = π(♦a)

In other words, we simply want π to a BAO-embedding, not justa boolean algebra embedding. The operator ♦σ defined on theprevious slide satisfies The Necessary Condition. But it is by nomeans clear that this is the only operator which satisfies it!

Introduction Varieties not closed under completions Overview of completions Conclusions

The Necessary Condition

. . . what exactly do we want to achieve?

Definition (Necessary Condition)

The Necessary Condition for a good operator f on Aσ: for everya ∈ A, f (π(a)) = π(♦a)

In other words, we simply want π to a BAO-embedding, not justa boolean algebra embedding. The operator ♦σ defined on theprevious slide satisfies The Necessary Condition. But it is by nomeans clear that this is the only operator which satisfies it!

Introduction Varieties not closed under completions Overview of completions Conclusions

Two notions of canonicity

This motivates the following two definitions

Definition (Jonsson-Tarski Canonicity)

A variety V of BAOs is Jonsson-Tarski canonical if for everyA ∈ V , Aσ with the operator ♦σ belongs to V

Definition (Abstract Canonicity)

A variety V is abstract canonical if for every every A ∈ V , thereis an operator f satisfying The Necessary Condition s.t. Aσ withthe operator f belongs to V

The second definition should be probably attributed to Chellasand/or Surendonk. It is obvious that JT-canonicity implies theabstract property. The converse, however, does not hold!

Introduction Varieties not closed under completions Overview of completions Conclusions

Two notions of canonicity

This motivates the following two definitions

Definition (Jonsson-Tarski Canonicity)

A variety V of BAOs is Jonsson-Tarski canonical if for everyA ∈ V , Aσ with the operator ♦σ belongs to V

Definition (Abstract Canonicity)

A variety V is abstract canonical if for every every A ∈ V , thereis an operator f satisfying The Necessary Condition s.t. Aσ withthe operator f belongs to V

The second definition should be probably attributed to Chellasand/or Surendonk. It is obvious that JT-canonicity implies theabstract property. The converse, however, does not hold!

Introduction Varieties not closed under completions Overview of completions Conclusions

Two notions of canonicity

This motivates the following two definitions

Definition (Jonsson-Tarski Canonicity)

A variety V of BAOs is Jonsson-Tarski canonical if for everyA ∈ V , Aσ with the operator ♦σ belongs to V

Definition (Abstract Canonicity)

A variety V is abstract canonical if for every every A ∈ V , thereis an operator f satisfying The Necessary Condition s.t. Aσ withthe operator f belongs to V

The second definition should be probably attributed to Chellasand/or Surendonk. It is obvious that JT-canonicity implies theabstract property. The converse, however, does not hold!

Introduction Varieties not closed under completions Overview of completions Conclusions

The McKinsey axiom and Heyting algebras

Theorem (Surendonk)

The variety of BAOs satisfying �♦x ≤ ♦�x is abstractcanonical

This variety, however, is not JT canonical, i.e., canonical in thestandard sense (Goldblatt)

Question

Is every variety of Heyting algebras abstract canonical? (sorryfor not defining explicitly this notion for non-boolean algebras)

Even if there is a counterexample, it is extremely hard to find.Notice the contrast with closure under MacN!

Introduction Varieties not closed under completions Overview of completions Conclusions

The McKinsey axiom and Heyting algebras

Theorem (Surendonk)

The variety of BAOs satisfying �♦x ≤ ♦�x is abstractcanonical

This variety, however, is not JT canonical, i.e., canonical in thestandard sense (Goldblatt)

Question

Is every variety of Heyting algebras abstract canonical? (sorryfor not defining explicitly this notion for non-boolean algebras)

Even if there is a counterexample, it is extremely hard to find.Notice the contrast with closure under MacN!

Introduction Varieties not closed under completions Overview of completions Conclusions

The McKinsey axiom and Heyting algebras

Theorem (Surendonk)

The variety of BAOs satisfying �♦x ≤ ♦�x is abstractcanonical

This variety, however, is not JT canonical, i.e., canonical in thestandard sense (Goldblatt)

Question

Is every variety of Heyting algebras abstract canonical? (sorryfor not defining explicitly this notion for non-boolean algebras)

Even if there is a counterexample, it is extremely hard to find.Notice the contrast with closure under MacN!

Introduction Varieties not closed under completions Overview of completions Conclusions

The McKinsey axiom and Heyting algebras

Theorem (Surendonk)

The variety of BAOs satisfying �♦x ≤ ♦�x is abstractcanonical

This variety, however, is not JT canonical, i.e., canonical in thestandard sense (Goldblatt)

Question

Is every variety of Heyting algebras abstract canonical? (sorryfor not defining explicitly this notion for non-boolean algebras)

Even if there is a counterexample, it is extremely hard to find.Notice the contrast with closure under MacN!

Introduction Varieties not closed under completions Overview of completions Conclusions

Advantages of JT canonicity

A natural question to ask then: why bother at all and definecanonicity Jonsson-Tarski style?

First reason: even this notion is pretty general!

Theorem (Gehrke, Harding, Venema)

Every MacN-closed variety is JT -canonical

Examples that the converse does not hold abound. Forexample, any variety of BAOs or Heyting algebrasHSP-generated by a finite algebra (corresponding to a logicdetermined by finite matrix) is canonical, though almost none isMacN-closed

Introduction Varieties not closed under completions Overview of completions Conclusions

Advantages of JT canonicity

A natural question to ask then: why bother at all and definecanonicity Jonsson-Tarski style?First reason: even this notion is pretty general!

Theorem (Gehrke, Harding, Venema)

Every MacN-closed variety is JT -canonical

Examples that the converse does not hold abound. Forexample, any variety of BAOs or Heyting algebrasHSP-generated by a finite algebra (corresponding to a logicdetermined by finite matrix) is canonical, though almost none isMacN-closed

Introduction Varieties not closed under completions Overview of completions Conclusions

Advantages of JT canonicity

A natural question to ask then: why bother at all and definecanonicity Jonsson-Tarski style?First reason: even this notion is pretty general!

Theorem (Gehrke, Harding, Venema)

Every MacN-closed variety is JT -canonical

Examples that the converse does not hold abound. Forexample, any variety of BAOs or Heyting algebrasHSP-generated by a finite algebra (corresponding to a logicdetermined by finite matrix) is canonical, though almost none isMacN-closed

Introduction Varieties not closed under completions Overview of completions Conclusions

Advantages of JT canonicity

A natural question to ask then: why bother at all and definecanonicity Jonsson-Tarski style?First reason: even this notion is pretty general!

Theorem (Gehrke, Harding, Venema)

Every MacN-closed variety is JT -canonical

Examples that the converse does not hold abound. Forexample, any variety of BAOs or Heyting algebrasHSP-generated by a finite algebra (corresponding to a logicdetermined by finite matrix) is canonical, though almost none isMacN-closed

Introduction Varieties not closed under completions Overview of completions Conclusions

Relational respresentation

Observe that as we defined ♦σX as a join of ♦σx for atomsbelows x , it is bound to be completely additive and recall thatdual algebras of Kripke frames were completely additive

Fact (essentially Jonsson and Tarski)

A BAO can be represented as a dual algebra of a Kripke frameiff it is complete, atomic and completely additive

I wrote essentially, as Jonsson and Tarski did not know Kripkeframes in 1951. This has been later extended by Thomason toa nice category-theoretical duality

Introduction Varieties not closed under completions Overview of completions Conclusions

Relational respresentation

Observe that as we defined ♦σX as a join of ♦σx for atomsbelows x , it is bound to be completely additive and recall thatdual algebras of Kripke frames were completely additive

Fact (essentially Jonsson and Tarski)

A BAO can be represented as a dual algebra of a Kripke frameiff it is complete, atomic and completely additive

I wrote essentially, as Jonsson and Tarski did not know Kripkeframes in 1951. This has been later extended by Thomason toa nice category-theoretical duality

Introduction Varieties not closed under completions Overview of completions Conclusions

Relational respresentation

Observe that as we defined ♦σX as a join of ♦σx for atomsbelows x , it is bound to be completely additive and recall thatdual algebras of Kripke frames were completely additive

Fact (essentially Jonsson and Tarski)

A BAO can be represented as a dual algebra of a Kripke frameiff it is complete, atomic and completely additive

I wrote essentially, as Jonsson and Tarski did not know Kripkeframes in 1951. This has been later extended by Thomason toa nice category-theoretical duality

Introduction Varieties not closed under completions Overview of completions Conclusions

Complexity

So, JT-canonicity implies another nice property:

Definition (Complexity)

A variety V is complex if for every A ∈ V , there is a Kripkeframe F s.t. A can be embedded in F+ and F+ ∈ V

Logicians know this property as strong Kripke completeness. Isthis notion equivalent to JT canonicity?

Introduction Varieties not closed under completions Overview of completions Conclusions

Complexity

So, JT-canonicity implies another nice property:

Definition (Complexity)

A variety V is complex if for every A ∈ V , there is a Kripkeframe F s.t. A can be embedded in F+ and F+ ∈ V

Logicians know this property as strong Kripke completeness. Isthis notion equivalent to JT canonicity?

Introduction Varieties not closed under completions Overview of completions Conclusions

Complexity

So, JT-canonicity implies another nice property:

Definition (Complexity)

A variety V is complex if for every A ∈ V , there is a Kripkeframe F s.t. A can be embedded in F+ and F+ ∈ V

Logicians know this property as strong Kripke completeness. Isthis notion equivalent to JT canonicity?

Introduction Varieties not closed under completions Overview of completions Conclusions

More counterexamples

AGAIN, NO!

Take R with strict order as a Kripke frame withboth ♦ and ♦P . The variety HSP-generated by it is complex butnot JT canonical (Wolter) As this example has residuatedoperators, it probably means that it is not abstract canonicaleither — but I have to check itAnd abstract canonicity does not imply complexity: theMcKinsey variety is not even complex (Wang) . . .Let us stop multiplying counterexamples here (for today) anddraw some morals

Introduction Varieties not closed under completions Overview of completions Conclusions

More counterexamples

AGAIN, NO! Take R with strict order as a Kripke frame withboth ♦ and ♦P . The variety HSP-generated by it is complex butnot JT canonical (Wolter)

As this example has residuatedoperators, it probably means that it is not abstract canonicaleither — but I have to check itAnd abstract canonicity does not imply complexity: theMcKinsey variety is not even complex (Wang) . . .Let us stop multiplying counterexamples here (for today) anddraw some morals

Introduction Varieties not closed under completions Overview of completions Conclusions

More counterexamples

AGAIN, NO! Take R with strict order as a Kripke frame withboth ♦ and ♦P . The variety HSP-generated by it is complex butnot JT canonical (Wolter) As this example has residuatedoperators, it probably means that it is not abstract canonicaleither — but I have to check it

And abstract canonicity does not imply complexity: theMcKinsey variety is not even complex (Wang) . . .Let us stop multiplying counterexamples here (for today) anddraw some morals

Introduction Varieties not closed under completions Overview of completions Conclusions

More counterexamples

AGAIN, NO! Take R with strict order as a Kripke frame withboth ♦ and ♦P . The variety HSP-generated by it is complex butnot JT canonical (Wolter) As this example has residuatedoperators, it probably means that it is not abstract canonicaleither — but I have to check itAnd abstract canonicity does not imply complexity: theMcKinsey variety is not even complex (Wang) . . .

Let us stop multiplying counterexamples here (for today) anddraw some morals

Introduction Varieties not closed under completions Overview of completions Conclusions

More counterexamples

AGAIN, NO! Take R with strict order as a Kripke frame withboth ♦ and ♦P . The variety HSP-generated by it is complex butnot JT canonical (Wolter) As this example has residuatedoperators, it probably means that it is not abstract canonicaleither — but I have to check itAnd abstract canonicity does not imply complexity: theMcKinsey variety is not even complex (Wang) . . .Let us stop multiplying counterexamples here (for today) anddraw some morals

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals

1 The notion of being closed under completions is verygeneral. Depending what exactly we want our completionsto do, we can focus on preservation of arbitrary joins,embeddability in the space of all prime filters (we can thinkabout them as non-standard points), having a relationalrepresentation . . . Each time we end up with a differentnotion. This is the first thing to remember

2 The second one: failure of being closed under anycompletion seems then to be a strong property and thusharder to prove than failure to be closed under someparticular kind of completions. But it is not exactly so

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals

1 The notion of being closed under completions is verygeneral. Depending what exactly we want our completionsto do, we can focus on preservation of arbitrary joins,embeddability in the space of all prime filters (we can thinkabout them as non-standard points), having a relationalrepresentation . . .

Each time we end up with a differentnotion. This is the first thing to remember

2 The second one: failure of being closed under anycompletion seems then to be a strong property and thusharder to prove than failure to be closed under someparticular kind of completions. But it is not exactly so

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals

1 The notion of being closed under completions is verygeneral. Depending what exactly we want our completionsto do, we can focus on preservation of arbitrary joins,embeddability in the space of all prime filters (we can thinkabout them as non-standard points), having a relationalrepresentation . . . Each time we end up with a differentnotion. This is the first thing to remember

2 The second one: failure of being closed under anycompletion seems then to be a strong property and thusharder to prove than failure to be closed under someparticular kind of completions. But it is not exactly so

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals

1 The notion of being closed under completions is verygeneral. Depending what exactly we want our completionsto do, we can focus on preservation of arbitrary joins,embeddability in the space of all prime filters (we can thinkabout them as non-standard points), having a relationalrepresentation . . . Each time we end up with a differentnotion. This is the first thing to remember

2 The second one: failure of being closed under anycompletion seems then to be a strong property and thusharder to prove than failure to be closed under someparticular kind of completions. But it is not exactly so

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals (continued)

In fact, even though it is a strong property, the proof that avariety is not closed under any completions can be quiteexplicit, as we saw. The reason: we abstract from irrelevantdetails concerning e.g., the structure of canonical extensionand focus on (possibly infinitary) arithmetical laws which haveto hold in all algebras from the variety. Yet another example of awell-known mathematical paradox:

A stronger theorem may be sometimes easier to prove

Thanks for patience and attention!

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals (continued)

In fact, even though it is a strong property, the proof that avariety is not closed under any completions can be quiteexplicit, as we saw. The reason: we abstract from irrelevantdetails concerning e.g., the structure of canonical extensionand focus on (possibly infinitary) arithmetical laws which haveto hold in all algebras from the variety. Yet another example of awell-known mathematical paradox:

A stronger theorem may be sometimes easier to prove

Thanks for patience and attention!

Introduction Varieties not closed under completions Overview of completions Conclusions

Morals (continued)

In fact, even though it is a strong property, the proof that avariety is not closed under any completions can be quiteexplicit, as we saw. The reason: we abstract from irrelevantdetails concerning e.g., the structure of canonical extensionand focus on (possibly infinitary) arithmetical laws which haveto hold in all algebras from the variety. Yet another example of awell-known mathematical paradox:

A stronger theorem may be sometimes easier to prove

Thanks for patience and attention!