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Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth

Epimorphism surjectivity and the Beth definability property

Tommaso Moraschini

Joint with: Guram Bezhanishvili and James Raftery

September 25, 2017

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Contents

• 1. Beth and epimorphisms
• 2. Blok-Hoogland’s conjecture
• 3. Finite depth

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Epimorphism surjectivity

Definition

Let K be a class of algebras. A homomorphism f : A → B in K is an epimorphism if for every pair g, h: B ⇒ C of homomorphisms in K if g ◦ f = h ◦ f , then g = h.

◮ Are epis surjective in a variety? ◮ Yes: Boolean algebras, Heyting algebras, lattices, semilattices

and (Abelian) groups.

◮ No: distributive lattices, rings with unity and monoids. ◮ Thus epimorphism surjectivity is not preserved in subvarieties!

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Beth property

◮ Let L be algebraizable with equivalence formulas ρ(x, y).

Definition

Let Γ be a set of formulas over X ∪ Z with X ∩ Z = ∅ and X = ∅.

• 1. Γ implicitly defines Z in terms of X if

Γ ∪ σ(Γ) ⊢L ρ(z, σz) for every z ∈ Z for every substitution σ that fixes X.

• 2. Γ explicitly defines Z in terms of X if for every z ∈ Z there is

a formula ϕz over X only such that Γ ⊢L ρ(z, ϕz).

◮ L has the (resp. finite) Beth property when 1 (resp. with Z

finite) implies 2.

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Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth

Transfer theorem

Definition

A homomorphism f : A → B is almost onto if B is generated by f (A) ∪ {b} for some b ∈ B.

Theorem (Blok and Hoogland)

Let L be an algebraizable logic.

• 1. L has the Beth property iff epis are surjective in Alg∗L.
• 2. L has the finite Beth property iff almost onto epis are

surjective in Alg∗L.

◮ Blok and Hoogland conjectured that

Beth property = finite Beth property.

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Why Heyting algebras?

◮ We want to establish Blok and Hoogland’s conjecture by

finding a variety (that algebraizes a logic) where:

• 1. Almost onto epimorphisms are surjective.
• 2. Epimorphisms need not be surjective.

Theorem (Kreisel)

Every axiomatic extension of IPC has the finite Beth property.

◮ This result can be re-stated as follows:

Theorem

In varieties of Heyting algebras almost onto epis are surjective.

◮ To establish Blok and Hoogland’s conjecture, it is enough to

find a variety of Heyting algebras where epis need not be surjective.

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K-epic subalgebras

Definition

Let K be a quasi-variety and B ∈ K. A subalgebra A ≤ B is K-epic if for every pair of homomorphisms f , g : B ⇒ C ∈ K if f ↾A= g ↾A , then f = g.

◮ Epis are surjective in K iff no B ∈ K has a proper K-epic

subalgebra.

Theorem (Campercholi)

Let K be a quasi-variety and A ≤ B ∈ K. TFAE:

• 1. A is a K-epic subalgebra of B.
• 2. For every b ∈ B there is a primitive positive formula ϕ(

x, y) and a ∈ A such that K ∀ x, y, z((ϕ( x, y)&ϕ( x, z)) → y ≈ z) and B ϕ( a, b).

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◮ Let A be the Heyting algebra depicted below and B the

subalgebra with universe {0, b0, b1, b2, . . . , 1}.

r r

• r

❅ ❅ r

• r

❅ ❅ r

• r

❅ ❅ r

• r

❅ ❅ r

• r

❅ ❅ r

• r

❅ ❅

• ❅

❅ ❅ 1 b0 c0 b1 c1 b2 c2

◮ We claim that B is a V(A)-epic subalgebra of A. ◮ We need to find primitive positive formulas that define partial

functions in V(A) and, moreover, construct A out of B.

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Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth

Partial functions

◮ Consider the conjunction of equations

ϕ(x0, x1, x2, y0, y1, y2) :=&

n≤2

(xn → yn ≈ yn &yn → xn ≈ xn)

&

n≤1

(xn ∧ yn ≈ xn+1 ∨ yn+1).

◮ and the primitive positive formula

Φ(x0, x1, x2, y0) := ∃y1y2ϕ.

◮ Φ define a partial 3-ary function in V(A): For every C ∈ V(A)

and a0, a1, a2 ∈ C there is at most one e ∈ C s.t. C Φ(a0, a1, a2, e).

◮ Applying this partial function to B we recover the whole A:

A Φ(bn+2, bn+1, bn, cn) for every n ∈ ω.

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Two Beth properties

◮ Epimorphisms need not to be surjective in V(A). ◮ Observe that V(A) satisfies the weak Pierce law

(y → x) ∨ (((x → y) → x) → x) ≈ 1.

◮ Then V(A) is locally finite.

Theorem (Blok-Hoogland’s conjecture)

• 1. Epimorphisms need not be surjective in locally finite varieties
• f Heyting algebras.
• 2. The Beth property and the finite Beth property are different in

locally tabular superintuitionistic logics.

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Rieger-Nishimura lattice

Definition

A Heyting algebra A has width n if the largest antichain in principal upsets of Pr(A), ⊆ has exactly n elements.

◮ Let Wn be the class of Heyting algebras of width ≤ n. It is a

variety.

◮ V(A) has width 2. ◮ The Rieger-Nishimura lattice has width 2.

Theorem

• 1. There is a continuum of varieties of Heyting algebras width

≤ 2 where epimorphisms need not be surjective.

• 2. Among them there is the variety generated by the

Rieger-Nishimura lattice.

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Finite depth

Definition

A Heyting algebra A has depth n if the longest chain in Pr(A), ⊆ has exactly n elements. Let HAn be the class of Heyting algebras of depth ≤ n.

Theorem (Maksimova and Ono)

HAn is a variety axiomatized by hn ≈ 1, where h0 = y and for n > 0 hn := xn ∨ (xn → hn−1).

◮ A variety of Heyting algebras has finite depth when its

members have finite depth.

Theorem

Let K be a variety of Heyting algberas. If K has finite depth, then epimorphisms are surjective in K.

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Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth

Consequences

◮ Finitely generated varieties of Heyting algebras are known to

have finite depth.

Corollary

• 1. Epimorphisms are surjective in finitely generated varieties of

Heyting algberas.

• 2. Tabular superintuitionistic logics have the Beth property.
• 3. Superintuitionistic logics, whose theorems include hn for some

n ∈ ω, have the Beth property.

• 4. Epimorphisms are surjective in all varieties of Gödel algebras.

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Strong epimorphism surjectivity

Definition

A class of algebras K has strong epimorphism surjectivity if whenever f : A → B is homomorphism in K and b ∈ B f (A), there are homomorphisms g, h: B ⇒ C in K such that g ◦ f = h ◦ f and g(b) = h(b).

Theorem (Maksimova)

There are finitely many varieties of Heyting algebras with strong epimorphism surjectitiy.

◮ There is a continuum of varieties of depth ≤ 3. ◮ Thus there is a continuum of varieties with epimorphism

surjectivity but not strong epimorphism surjectivity.

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Thanks for coming!

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