SLIDE 1 Epimorphisms in Varieties
JAMES RAFTERY
(Univ. Pretoria, South Africa)
JOINT WORK WITH
Guram Bezhanishvili (New Mexico State Univ., USA) Tommaso Moraschini (Acad. Sci., Czech Republic)
SLIDE 2 In a concrete category K, a morphism h: A − → B is called a (K-) epimorphism when, for any K-morphisms f, g : B − → C, if f ◦ h = g ◦ h, then f = g.
✐
A
✲ ✒✑ ✓✏ ✐
B C ∈ K
✒✑ ✓✏ ✲ ✲
g f h
✑ ✑
h[A]
Note: (1) Surjective K-morphisms are K-epimorphisms. (2) We say that K has the ES property if all K-epimorphisms are surjective. (3) A variety K has the ES property iff no B ∈ K has a (K-) epic (proper) subalgebra D, i.e., one such that any K-morphism f : B − → C is determined by f|D.
SLIDE 3 In a concrete category K, a morphism h: A − → B is called a (K-) epimorphism when, for any K-morphisms f, g : B − → C, if f ◦ h = g ◦ h, then f = g.
✐
A
✲ ✒✑ ✓✏ ✐
B C ∈ K
✒✑ ✓✏ ✲ ✲
g f h
✑ ✑
h[A]
Note: (1) Surjective K-morphisms are K-epimorphisms. (2) We say that K has the ES property if all K-epimorphisms are surjective. (3) A variety K has the ES property iff no B ∈ K has a (K-) epic (proper) subalgebra D, i.e., one such that any K-morphism f : B − → C is determined by f|D.
SLIDE 4 In a concrete category K, a morphism h: A − → B is called a (K-) epimorphism when, for any K-morphisms f, g : B − → C, if f ◦ h = g ◦ h, then f = g.
✐
A
✲ ✒✑ ✓✏ ✐
B C ∈ K
✒✑ ✓✏ ✲ ✲
g f h
✑ ✑
h[A]
Note: (1) Surjective K-morphisms are K-epimorphisms. (2) We say that K has the ES property if all K-epimorphisms are surjective. (3) A variety K has the ES property iff no B ∈ K has a (K-) epic (proper) subalgebra D, i.e., one such that any K-morphism f : B − → C is determined by f|D.
SLIDE 5 In a concrete category K, a morphism h: A − → B is called a (K-) epimorphism when, for any K-morphisms f, g : B − → C, if f ◦ h = g ◦ h, then f = g.
✐
A
✲ ✒✑ ✓✏ ✐
B C ∈ K
✒✑ ✓✏ ✲ ✲
g f h
✑ ✑
h[A]
Note: (1) Surjective K-morphisms are K-epimorphisms. (2) We say that K has the ES property if all K-epimorphisms are surjective. (3) A variety K has the ES property iff no B ∈ K has a (K-) epic (proper) subalgebra D, i.e., one such that any K-morphism f : B − → C is determined by f|D.
SLIDE 6 In a concrete category K, a morphism h: A − → B is called a (K-) epimorphism when, for any K-morphisms f, g : B − → C, if f ◦ h = g ◦ h, then f = g.
✐
A
✲ ✒✑ ✓✏ ✐
B C ∈ K
✒✑ ✓✏ ✲ ✲
g f h
✑ ✑
h[A]
Note: (1) Surjective K-morphisms are K-epimorphisms. (2) We say that K has the ES property if all K-epimorphisms are surjective. (3) A variety K has the ES property iff no B ∈ K has a (K-) epic (proper) subalgebra D, i.e., one such that any K-morphism f : B − → C is determined by f|D.
SLIDE 7 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 8 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 9 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 10 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 11 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 12 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 13 (4) {Groups}, {R-modules}, {Lattices}, {Semilattices} and {Boolean algebras} are varieties with the ES property. (5) The variety {Rings} lacks ES, as Z is epic in Q. This is because, although multiplicative inverses needn’t exist, they are implicitly definable in rings—i.e., uniquely determined or non-existent. (6) The ES property needn’t persist in subvarieties: it holds in {Lattices}, but not in {Distributive Lattices}, where
s s s
is epic in
s
❅ ❅ s
❅ ❅
This is due to the uniqueness of existent complements in distributive lattices.
SLIDE 14 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 15 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 16 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 17 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 18 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 19 Why study ES? Let K be a variety algebraizing a logic ⊢, e.g., {Boolean algebras} ← → classical propositional logic,
→ intuitionistic propositional logic.
- Theorem. (Blok & Hoogland, 2006) K has the ES property iff
⊢ has the infinite Beth (definability) property, which means: whenever Γ ⊆ Form(X ˙ ∪ Z) and Γ ∪ h[Γ] ⊢ z ↔ h(z) for all z ∈ Z and all substitutions h (of formulas for variables) such that h(x) = x for all x ∈ X, THEN for each z ∈ Z, there’s a formula ϕz ∈ Form(X) such that Γ ⊢ z ↔ ϕz.
SLIDE 20 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 21 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 22 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 23 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 24 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 25 The finite Beth property makes the same demand, but only when Z is finite.
emeti, 1984) ⊢ has the finite Beth property iff K has the weak ES property, which means: every ‘almost onto’ K-epimorphism is onto, where ‘h: A − → B is almost onto’ means that B is generated by h[A] ∪ {b} for some b ∈ B.
- Problem. Does the finite Beth property imply the infinite one?
Blok-Hoogland Conjecture: No.
SLIDE 26 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 27 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 28 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 29 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 30 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 31 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 32 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 33 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 34 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 35 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 36 In algebraic terms:
- Question. Does weak ES imply ES (at least for varieties)?
Yes, for amalgamable varieties (known), so we eschew these. Where to look? Although {Boolean algebras} have ES, the 2ℵ0 varieties of Heyting algebras ALL have weak ES (Kreisel, 1960), but only finitely many of them are amalgamable (Maksimova, 1970s). HA := {all Heyting algebras} has ES.
- Question. Which subvarieties of HA have ES?
- Answer. Not all. (Blok-Hoogland Conjecture confirmed.)
Some of the counter-examples are locally finite.
SLIDE 37 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 38 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 39 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 40 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 41 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 42 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 43 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 44 NEW POSITIVE RESULTS
- Theorem. If a variety of Heyting algebras has finite depth, then
it has surjective epimorphisms. (2ℵ0 examples.) [Known: finitely generated ⇒ finite depth ⇒ locally finite.]
- Corollary. Every finitely generated variety of Heyting algebras
has surjective epimorphisms. [In contrast, it’s known that only finitely many subvarieties of HA have the so-called strong ES property: whenever A B ∈ K and b ∈ B\A, there are two K-morphisms f, g : B − → C that agree on A but not at b (Maksimova, 2000).]
- Corollary. Every variety of G¨
- del algebras (i.e., of subdirect
products of totally ordered Heyting algebras) has ES.
SLIDE 45 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 46 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 47 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 48 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 49 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 50 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 51 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 52 Everything said thus far applies equally to Brouwerian algebras, i.e., to possibly unbounded Heyting algebras. Logical Interpretation:
- Theorem. If a super-intuitionistic [or positive] logic is
tabular—or more generally if its theorems include a formula from the sequence h0 := y; hn := xn ∨ (xn → hn−1) (0 < n ∈ ω), then it has the infinite Beth property. Likewise all G¨
Even the finite Beth property fails in all axiomatic extensions
- f Hajek’s Basic Logic (BL), excepting the G¨
- del logics
[Montagna, 2006]. Likewise many relevance logics [Urquhart, 1999], but new exceptions emerge here.
SLIDE 53 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 54 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 55 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 56 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 57 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 58 Beyond Heyting/Brouwerian/BL algebras More general than Heyting/BL algebras are residuated lattices A = A; ·, →, ∧, ∨, e. [A; ∧, ∨ is a lattice and A; ·, e a commutative monoid with x · y z ⇐ ⇒ y x → z (law of residuation).] Several varieties of these are categorically equivalent to varieties of (enriched) G¨
- del algebras [Galatos & R, 2012/15].
The ES property is categorical, so it transfers. With more work, we obtain:
SLIDE 59
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 60
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 61
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 62
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 63
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 64
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 65
- Theorem. Every variety of Sugihara monoids has ES.
[A Sugihara monoid A = A; ·, →, ∧, ∨, ¬, e is a residuated distributive lattice with an involution ¬, where · is idempotent. It needn’t be integral, i.e., e needn’t be its top element.] The lattice of varieties of Sugihara monoids is denumerable, but not a chain.
- Corollary. Every axiomatic extension of the relevance logic
RMt has the infinite Beth property.
SLIDE 66
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 67
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 68
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 69
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 70
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 71
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 72
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 73
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 74
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 75
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 76
The proof of ES for varieties of Heyting algebras A = A; →, ∧, ∨, ⊤, ⊥ of finite depth uses Esakia duality. From A, we construct an Esakia space A∗ := Pr A; ⊆, τ. Pr A is the set of all prime filters of A (i.e., all lattice filters F with ⊤ ∈ F, such that A\F is closed under ∨), and τ is a certain topology on Pr A. For a ∈ A, we define ϕ(a) = {F ∈ Pr A : a ∈ F} and ϕ(a)c = {F ∈ Pr A : a / ∈ F}. A sub-basis for τ is then {ϕ(a) : a ∈ A} ∪ {ϕ(a)c : a ∈ A}. For a HA–morphism h: A − → B, define h∗ : B∗ − → A∗ by F → h−1[F].
SLIDE 77
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 78
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 79
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 80
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 81
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 82
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 83
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 84
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 85
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 86
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 87
- Theorem. [Esakia, 1974] A duality between HA and the
category ESP of Esakia spaces (and morphisms) is established by the functor A → A∗ ; h → h∗. I.e., the categories HA and ESPop are equivalent. In general, an Esakia space X = X; , τ comprises a po-set X; and a compact Hausdorff topology τ on X in which every open set is a union of clopen sets; ↑x is closed, for all x ∈ X; ↓W is clopen, for all clopen W ⊆ X. An Esakia morphism h: X − → Y between such spaces is a continuous function such that h[ ↑x] = ↑h(x), for all x ∈ X. The reverse functor X → X ∗ ∈ HA; h → h∗ sends X to its set
- f clopen up-sets (including X and ∅), equipped with operations
∩, ∪ and U → V := X\ ↓(U\V), while h∗ : U → h−1[U].
SLIDE 88 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 89 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 90 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 91 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 92 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 93 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 94 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 95 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 96 If K is a subvariety of HA, then (−)∗ and (−)∗ restrict to a duality between K and K∗ := I{A∗ : A ∈ K} ⊆ ESP. Depth: Let A be a Heyting algebra, with dual A∗ = Pr A; ⊆, τ. We say that A (and A∗) have depth n ∈ ω if, in A∗, there’s a chain p1 < . . . < pn, but no chain q1 < . . . < qn+1. Depths of elements of A∗ are defined similarly. We say that K ⊆ HA has depth n if all A ∈ K do.
- Fact. HAn := {A ∈ HA : depth(A) n} is a variety, ∀n ∈ ω.
[Ideas from Hosoi 1967; Ono 1970; Maksimova 1972.] HA0 = {trivials}; HA1 = {Boolean algebras}; HA3 already has 2ℵ0 subvarieties [Kuznetsov 1974].
SLIDE 97
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 98
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 99
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 100
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 101
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 102
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 103
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 104
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 105
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 106
- Theorem. Let K ⊆ HA be a variety of finite depth, n say.
Then K has surjective epimorphisms. Proof sketch. First, K has ES iff all K∗-monomorphisms h are injective. [Here, h ◦ f = h ◦ g = ⇒ f = g.] We induct on n, the case n = 0 being trivial. Let n > 0. W.l.o.g., we can restrict to the following situation, in which h: X − → Y is a K∗-mono, with x = y in X, where X = ↑{x, y} and — with a view to contradiction — h(x) = h(y).
X Y P r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y h ✲ h(x) =h(y) r ✫✪ ✬✩
Here, P := {u ∈ X : depth(u) < n}. By the induction hypothesis, h|P is one-to-one, so x or y has depth = n.
SLIDE 107 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 108 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 109 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 110 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 111 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 112 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 113 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 114 Case: x, y both have depth n. (The other case is easier.) As h is an ESP-morphism and h|P is one-to-one, we can show that x and y have the same covers in X. It follows that ↑x and ↑y are isomorphic Esakia spaces. Let W be the disjoint union of ↑x, ↑y and a copy ↑z of ↑x.
W X P r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ r r r r r r r r r r ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ax ay az a x y z g1 ✲ ✲ g2 r r r . . . . . .
❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ r r x y
Each strict upper bound a of x in X yields copies ax > x, ay > y and az > z of itself in W. Sending these back to a, we get Esakia morphisms g1, g2 : W − → X differing only in that g1 : z → x, while g2 : z → y (both: x → x; y → y).
SLIDE 115
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 116
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 117
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 118
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 119
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 120
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 121
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 122
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 123
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 124
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 125
Now h ◦ g1 = h ◦ g2 : W − → Y ∈ K∗ (as h(x) = h(y)). Since g1 = g2, this will contradict the fact that h is a K∗-monomorphism, provided that W ∈ K∗. As ↑x is a closed up-set of X, the inclusion i : ( ↑x) − → X is an ESP-morphism, so i∗ : X ∗ − → ( ↑x)∗ is onto, i.e., ( ↑x)∗ ∈ H(X ∗) ⊆ H(K) ⊆ K (since K∗ is a variety). So, ( ↑x)∗, ( ↑y)∗, ( ↑z)∗ ∈ K. So, A := ( ↑x)∗ × ( ↑y)∗ × ( ↑z)∗ ∈ P(K) ⊆ K. As it happens, A∗ ∼ = W := ( ↑x) ˙ ∪ ( ↑y) ˙ ∪ ( ↑z), so W ∈ K∗, as required.
SLIDE 126 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 127 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 128 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 129 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 130 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 131 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 132 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 133 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
SLIDE 134 A proper epic subalgebra in a Heyting algebra variety
s s
❅ ❅ s
❅ ❅ s
❅ ❅ s
❅ ❅ s
s
❅ ❅
❅ ❅ s⊥ ⊤ b0 c0 b1 c1 b2 c2 The variety V(A) generated by the Heyting algebra A on the left lacks the ES property, confirming the Blok-Hoogland conjecture. The red elements form a V(A)-epic subalgebra. V(A) is locally finite and has a fairly simple finite axiomatization. An explicit failure of the infinite Beth property can be extracted from this example. In the finitely subdirectly irreducible (but not all) members of V(A), the ‘incomparable companion’
- f an element is implicitly definable, but not
explicitly.
