Uniform Interpolation Part II: An Algebraic Framework George - - PowerPoint PPT Presentation

Uniform Interpolation

Part II: An Algebraic Framework George Metcalfe

Mathematical Institute University of Bern BLAST 2018, University of Denver, 6-10 August 2018

George Metcalfe (University of Bern) Uniform Interpolation August 2018 1 / 30

Yesterday

Theorem (Pitts 1992)

For any formula α(x, y) of intuitionistic propositional logic IL, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), α(x, y) ⊢IL β(y, z) ⇐ ⇒ αR(y) ⊢IL β(y, z) β(y, z) ⊢IL α(x, y) ⇐ ⇒ β(y, z) ⊢IL αL(y).

A.M. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57 (1992), 33–52.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 2 / 30

Remarks from Yesterday

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30

Remarks from Yesterday

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent intermediate logics that admit Craig interpolation (Maksimova 1977),

George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30

Remarks from Yesterday

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent intermediate logics that admit Craig interpolation (Maksimova 1977), and all of these also have uniform interpolation (Ghilardi and Zawadowski 2002).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30

Remarks from Yesterday

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent intermediate logics that admit Craig interpolation (Maksimova 1977), and all of these also have uniform interpolation (Ghilardi and Zawadowski 2002). Iemhoff has shown recently that any logic admitting a certain Dyckhoff-style decomposition calculus has uniform interpolation.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30

Today

What does (uniform) interpolation mean algebraically?

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Equations and Equational Classes

Let Tm(x) denote the term algebra of an algebraic language L with at least one constant over a set of variables x.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Equations and Equational Classes

Let Tm(x) denote the term algebra of an algebraic language L with at least one constant over a set of variables x. An L-equation is an ordered pair s, t of L-terms, also written s ≈ t.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Equations and Equational Classes

Let Tm(x) denote the term algebra of an algebraic language L with at least one constant over a set of variables x. An L-equation is an ordered pair s, t of L-terms, also written s ≈ t. We let V be any variety defined by L-equations, e.g., Boolean algebras, Heyting algebras, MV-algebras, modal algebras, bounded lattices, groups. . .

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Equational Consequence

For any set of L-equations Σ ∪ {ε} with variables in x, we write Σ | =V ε

George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30

Equational Consequence

For any set of L-equations Σ ∪ {ε} with variables in x, we write Σ | =V ε if for any A ∈ V and homomorphism e : Tm(x) → A,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30

Equational Consequence

For any set of L-equations Σ ∪ {ε} with variables in x, we write Σ | =V ε if for any A ∈ V and homomorphism e : Tm(x) → A, Σ ⊆ ker(e) = ⇒ ε ∈ ker(e), where ker(e) := {s, t | e(s) = e(t)} is the kernel of e.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30

Equational Consequence

For any set of L-equations Σ ∪ {ε} with variables in x, we write Σ | =V ε if for any A ∈ V and homomorphism e : Tm(x) → A, Σ ⊆ ker(e) = ⇒ ε ∈ ker(e), where ker(e) := {s, t | e(s) = e(t)} is the kernel of e. We also write Σ | =V ∆ to denote that Σ | =V ε for all ε ∈ ∆.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30

Deductive Interpolation

V admits deductive interpolation if whenever Σ(x, y) | =V ε(y, z), there exists a set of equations ∆(y) such that Σ(x, y) | =V ∆(y) and ∆(y) | =V ε(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30

Deductive Interpolation

V admits deductive interpolation if whenever Σ(x, y) | =V ε(y, z), there exists a set of equations ∆(y) such that Σ(x, y) | =V ∆(y) and ∆(y) | =V ε(y, z). Equivalently, V admits deductive interpolation if for any set of equations Σ(x, y), there exists a set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30

Deductive Interpolation

V admits deductive interpolation if whenever Σ(x, y) | =V ε(y, z), there exists a set of equations ∆(y) such that Σ(x, y) | =V ∆(y) and ∆(y) | =V ε(y, z). Equivalently, V admits deductive interpolation if for any set of equations Σ(x, y), there exists a set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z). (Just define ∆(y) := {ε(y) | Σ(x, y) | =V ε(y)}.)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30

Congruences

A congruence Θ on an algebra A is an equivalence relation satisfying a1, b1, . . . , an, bn ∈ Θ = ⇒ ⋆(a1, . . . , an), ⋆(b1, . . . , bn) ∈ Θ for every n-ary operation ⋆ of A.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Congruences

A congruence Θ on an algebra A is an equivalence relation satisfying a1, b1, . . . , an, bn ∈ Θ = ⇒ ⋆(a1, . . . , an), ⋆(b1, . . . , bn) ∈ Θ for every n-ary operation ⋆ of A. The congruences of A always form a complete lattice Con A.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Free Algebras

The free algebra of V over a set of variables x can be defined as F(x) = Tm(x)/ΘV where s ΘV t :⇐ ⇒ V | = s ≈ t.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 30

Free Algebras

The free algebra of V over a set of variables x can be defined as F(x) = Tm(x)/ΘV where s ΘV t :⇐ ⇒ V | = s ≈ t. We write t to denote both a term t in Tm(x) and [t] in F(x).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 30

Equational Consequence Again

Lemma

For any set of equations Σ ∪ {s ≈ t} with variables in x, Σ | =V s ≈ t ⇐ ⇒ s, t ∈ CgF(x)(Σ), where CgF(x)(Σ) is the congruence on F(x) generated by Σ.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Lifting Inclusions

The inclusion map i : F(y) → F(x, y)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Lifting Inclusions

The inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ])

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Lifting Inclusions

The inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ]) i−1 : Con F(x, y) → Con F(y); Ψ → i−1[Ψ] = Ψ ∩ F(y)2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Lifting Inclusions

The inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ]) i−1 : Con F(x, y) → Con F(y); Ψ → i−1[Ψ] = Ψ ∩ F(y)2. Note that the pair i∗, i−1 is an adjunction, i.e., i∗(Θ) ⊆ Ψ ⇐ ⇒ Θ ⊆ i−1(Ψ).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Deductive Interpolation Again

The following are equivalent: (1) V admits deductive interpolation, i.e., for any set of equations Σ(x, y), there exists a set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Deductive Interpolation Again

The following are equivalent: (1) V admits deductive interpolation, i.e., for any set of equations Σ(x, y), there exists a set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z). (2) For any finite sets x, y, z, the following diagram commutes: Con F(x, y) Con F(y) Con F(x, y, z) Con F(y, z) i−1 j∗ k−1 l∗ where i, j, k, and l denote inclusion maps between free algebras.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

The Amalgamation Property

B1

A

1 2

B2

V admits the amalgamation property if for any A, B1, B2 ∈ V, and embeddings σ1 : A → B1, σ2 : A → B2,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

The Amalgamation Property

B1

C A

1 2

B2

V admits the amalgamation property if for any A, B1, B2 ∈ V, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ V

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

The Amalgamation Property

B1

C A

1 !1 2 !2

B2

V admits the amalgamation property if for any A, B1, B2 ∈ V, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ V and embeddings τ1 : B1 → C and τ2 : B2 → C

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

The Amalgamation Property

B1

C A

1 !1 2 !2

B2

V admits the amalgamation property if for any A, B1, B2 ∈ V, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ V and embeddings τ1 : B1 → C and τ2 : B2 → C such that τ1σ1 = τ2σ2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

A Key Lemma

Lemma (Pigozzi 1972)

V admits the amalgamation property if and only if for any disjoint sets x, y, z and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfying Θ ∩ F(y)2 = Ψ ∩ F(y)2,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Key Lemma

Lemma (Pigozzi 1972)

V admits the amalgamation property if and only if for any disjoint sets x, y, z and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfying Θ ∩ F(y)2 = Ψ ∩ F(y)2, there exists Φ ∈ Con F(x, y, z) satisfying Θ = Φ ∩ F(x, y)2 and Ψ = Φ ∩ F(y, z)2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Key Lemma

Lemma (Pigozzi 1972)

V admits the amalgamation property if and only if for any disjoint sets x, y, z and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfying Θ ∩ F(y)2 = Ψ ∩ F(y)2, there exists Φ ∈ Con F(x, y, z) satisfying Θ = Φ ∩ F(x, y)2 and Ψ = Φ ∩ F(y, z)2. This property of congruences of free algebras can be reformulated in terms

• f consequence as the so-called Robinson property.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

Proof Sketch (⇒)

Suppose that V admits the amalgamation property and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfy Φ0 := Θ ∩ F(y)2 = Ψ ∩ F(y)2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Proof Sketch (⇒)

Suppose that V admits the amalgamation property and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfy Φ0 := Θ ∩ F(y)2 = Ψ ∩ F(y)2. We define A = F(y)/Φ0, B = F(x, y)/Θ, and C = F(y, z)/Ψ, F(x, y) F(x, y, z) F(y) F(y, z) B D A C

g

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Proof Sketch (⇒)

Suppose that V admits the amalgamation property and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfy Φ0 := Θ ∩ F(y)2 = Ψ ∩ F(y)2. We define A = F(y)/Φ0, B = F(x, y)/Θ, and C = F(y, z)/Ψ, yielding an amalgam D F(x, y) F(x, y, z) F(y) F(y, z) B D A C

g

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Proof Sketch (⇒)

Suppose that V admits the amalgamation property and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfy Φ0 := Θ ∩ F(y)2 = Ψ ∩ F(y)2. We define A = F(y)/Φ0, B = F(x, y)/Θ, and C = F(y, z)/Ψ, yielding an amalgam D and a surjective homomorphism g : F(x, y, z) → D F(x, y) F(x, y, z) F(y) F(y, z) B D A C

g

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Proof Sketch (⇒)

Suppose that V admits the amalgamation property and Θ ∈ Con F(x, y), Ψ ∈ Con F(y, z) satisfy Φ0 := Θ ∩ F(y)2 = Ψ ∩ F(y)2. We define A = F(y)/Φ0, B = F(x, y)/Θ, and C = F(y, z)/Ψ, yielding an amalgam D and a surjective homomorphism g : F(x, y, z) → D with Φ := ker(g) satisfying Θ = Φ ∩ F(x, y)2 and Ψ = Φ ∩ F(y, z)2. F(x, y) F(x, y, z) F(y) F(y, z) B D A C

g

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Proof Sketch (⇐)

Let B, C ∈ V be generated by x, y and y, z, respectively, with a common subalgebra A generated by y.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Proof Sketch (⇐)

Let B, C ∈ V be generated by x, y and y, z, respectively, with a common subalgebra A generated by y. Consider the surjective homomorphisms πA : F(y) → A, πB : F(x, y) → B, and πC : F(y, z) → C. F(x, y) F(x, y, z) F(y) F(y, z) B D A C

πB πA πC

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Proof Sketch (⇐)

Let B, C ∈ V be generated by x, y and y, z, respectively, with a common subalgebra A generated by y. Consider the surjective homomorphisms πA : F(y) → A, πB : F(x, y) → B, and πC : F(y, z) → C. Then Θ = ker(πB), Ψ = ker(πC) satisfy Θ ∩ FV(y)2 = Ψ ∩ F(y)2 F(x, y) F(x, y, z) F(y) F(y, z) B D A C

πB πA πC

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Proof Sketch (⇐)

Let B, C ∈ V be generated by x, y and y, z, respectively, with a common subalgebra A generated by y. Consider the surjective homomorphisms πA : F(y) → A, πB : F(x, y) → B, and πC : F(y, z) → C. Then Θ = ker(πB), Ψ = ker(πC) satisfy Θ ∩ FV(y)2 = Ψ ∩ F(y)2 so, by assumption, there exists Φ ∈ Con F(x, y, z) such that Φ ∩ F(x, y)2 = Θ and Φ ∩ F(y, z)2 = Ψ. F(x, y) F(x, y, z) F(y) F(y, z) B D A C

πB πA πC

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Proof Sketch (⇐)

Let B, C ∈ V be generated by x, y and y, z, respectively, with a common subalgebra A generated by y. Consider the surjective homomorphisms πA : F(y) → A, πB : F(x, y) → B, and πC : F(y, z) → C. Then Θ = ker(πB), Ψ = ker(πC) satisfy Θ ∩ FV(y)2 = Ψ ∩ F(y)2 so, by assumption, there exists Φ ∈ Con F(x, y, z) such that Φ ∩ F(x, y)2 = Θ and Φ ∩ F(y, z)2 = Ψ. The required amalgam is then D = F(x, y, y)/Φ. F(x, y) F(x, y, z) F(y) F(y, z) B D A C

πB πA πC

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

A Bridge Theorem

Theorem (Pigozzi, Bacsich, Maksimova, Czelakowski,. . . )

A variety with the congruence extension property admits the deductive interpolation property if and only if it admits the amalgamation property.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Further Relationships. . .

CEP + FAP m TIP = ) AP = ) WAP = ) FAP m m m m MIP = ) RP = ) CDIP = ) DIP m DIP + EP

• G. Metcalfe, F. Montagna, and C. Tsinakis.

Amalgamation and interpolation in ordered algebras. Journal of Algebra, 402:21–82, 2014.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

But Now. . .

Can we describe uniform interpolation algebraically?

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Deductive Interpolation

V has deductive interpolation if for any set of equations Σ(x, y), there exists a set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

Right Uniform Deductive Interpolation

V has right uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

Right Uniform Deductive Interpolation

V has right uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z). Equivalently, V has deductive interpolation and for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

Right Uniform Deductive Interpolation

V has right uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y, z) ⇐ ⇒ ∆(y) | =V ε(y, z). Equivalently, V has deductive interpolation and for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y). But what does this extra ingredient mean algebraically?

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

Compact Congruences

The compact (equivalently, finitely generated) congruences of an algebra A always form a join-semilattice KCon A.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Compact Congruences

The compact (equivalently, finitely generated) congruences of an algebra A always form a join-semilattice KCon A. Recall that the inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ]) i−1 : Con F(x, y) → Con F(y); Ψ → i−1[Ψ] = Ψ ∩ F(y)2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Compact Congruences

The compact (equivalently, finitely generated) congruences of an algebra A always form a join-semilattice KCon A. Recall that the inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ]) i−1 : Con F(x, y) → Con F(y); Ψ → i−1[Ψ] = Ψ ∩ F(y)2. The compact lifting of i restricts i∗ to KCon F(y) → KCon F(x, y);

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Compact Congruences

The compact (equivalently, finitely generated) congruences of an algebra A always form a join-semilattice KCon A. Recall that the inclusion map i : F(y) → F(x, y) “lifts” to the maps i∗ : Con F(y) → Con F(x, y); Θ → CgF(x,y)(i[Θ]) i−1 : Con F(x, y) → Con F(y); Ψ → i−1[Ψ] = Ψ ∩ F(y)2. The compact lifting of i restricts i∗ to KCon F(y) → KCon F(x, y); it has a right adjoint if i−1 restricts to KCon F(x, y) → KCon F(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Finitely Generated and Finitely Presented Algebras

An algebra A ∈ V is called finitely generated if it is generated by a finite subset of A;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

Finitely Generated and Finitely Presented Algebras

An algebra A ∈ V is called finitely generated if it is generated by a finite subset of A; finitely presented if it is isomorphic to F(x)/Θ for some finite set x and compact congruence Θ on F(x).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

Finitely Generated and Finitely Presented Algebras

An algebra A ∈ V is called finitely generated if it is generated by a finite subset of A; finitely presented if it is isomorphic to F(x)/Θ for some finite set x and compact congruence Θ on F(x).

Useful Lemma

If A ∈ V is finitely presented and isomorphic to F(x)/Θ for some finite set x and congruence Θ on F(x), then Θ is compact.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

Coherence

The notion of coherence originated in sheaf theory and has been studied quite widely in algebra, e.g., in connection with groups, rings, and monoids.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Coherence

The notion of coherence originated in sheaf theory and has been studied quite widely in algebra, e.g., in connection with groups, rings, and monoids. Following Wheeler, a variety V is coherent if all finitely generated subalgebras of finitely presented members of V are finitely presented.

W.H. Wheeler. Model-companions and definability in existentially complete structures. Israel Journal of Mathematics 25 (1976), 305–330.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Coherence

The notion of coherence originated in sheaf theory and has been studied quite widely in algebra, e.g., in connection with groups, rings, and monoids. Following Wheeler, a variety V is coherent if all finitely generated subalgebras of finitely presented members of V are finitely presented.

W.H. Wheeler. Model-companions and definability in existentially complete structures. Israel Journal of Mathematics 25 (1976), 305–330.

Note that clearly every locally finite variety is coherent.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Coherence

The notion of coherence originated in sheaf theory and has been studied quite widely in algebra, e.g., in connection with groups, rings, and monoids. Following Wheeler, a variety V is coherent if all finitely generated subalgebras of finitely presented members of V are finitely presented.

W.H. Wheeler. Model-companions and definability in existentially complete structures. Israel Journal of Mathematics 25 (1976), 305–330.

Note that clearly every locally finite variety is coherent. (Homework question. Is your favourite variety coherent?)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

The Missing Ingredient

Theorem (Kowalski and Metcalfe 2017)

The following are equivalent: (1) For any finite set of equations Σ(x, y), there is a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

The Missing Ingredient

Theorem (Kowalski and Metcalfe 2017)

The following are equivalent: (1) For any finite set of equations Σ(x, y), there is a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y). (2) For finite x, y, the compact lifting of F(y) ֒ → F(x, y) has a right adjoint; that is, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

The Missing Ingredient

Theorem (Kowalski and Metcalfe 2017)

The following are equivalent: (1) For any finite set of equations Σ(x, y), there is a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y). (2) For finite x, y, the compact lifting of F(y) ֒ → F(x, y) has a right adjoint; that is, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (3) V is coherent.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

The Missing Ingredient

Theorem (Kowalski and Metcalfe 2017)

The following are equivalent: (1) For any finite set of equations Σ(x, y), there is a finite set of equations ∆(y) such that Σ(x, y) | =V ε(y) ⇐ ⇒ ∆(y) | =V ε(y). (2) For finite x, y, the compact lifting of F(y) ֒ → F(x, y) has a right adjoint; that is, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (3) V is coherent.

Corollary (Pitts 1992)

The variety of Heyting algebras is coherent.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V. Let x, y and y be finite sets generating A and B, respectively.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V. Let x, y and y be finite sets generating A and B, respectively. The onto homomorphism h: F(x, y) → A restricts to k : F(y) → B, which is also

• nto.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V. Let x, y and y be finite sets generating A and B, respectively. The onto homomorphism h: F(x, y) → A restricts to k : F(y) → B, which is also

• nto. But ker h ∈ KCon F(x, y) by the useful lemma,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V. Let x, y and y be finite sets generating A and B, respectively. The onto homomorphism h: F(x, y) → A restricts to k : F(y) → B, which is also

• nto. But ker h ∈ KCon F(x, y) by the useful lemma, so by assumption,

ker k = ker h ∩ F(y)2 ∈ KCon F(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof of (2) ⇔ (3)

Proof.

We prove that V is coherent if and only if for any finite x, y, Θ ∈ KCon F(x, y) = ⇒ Θ ∩ F(y)2 ∈ KCon F(y). (⇒) Let V be coherent and consider finite x, y and Θ ∈ KCon F(x, y). F(x, y)/Θ is finitely presented and, by coherence, so is F(y)/(Θ ∩ F(y)2). Hence, by the useful lemma, Θ ∩ F(y)2 ∈ KCon F(y). (⇐) Let B be a finitely generated subalgebra of a finitely presented A ∈ V. Let x, y and y be finite sets generating A and B, respectively. The onto homomorphism h: F(x, y) → A restricts to k : F(y) → B, which is also

• nto. But ker h ∈ KCon F(x, y) by the useful lemma, so by assumption,

ker k = ker h ∩ F(y)2 ∈ KCon F(y). Hence B is finitely presented.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Another Bridge Theorem

Theorem (Kowalski and Metcalfe 2017)

A variety with the congruence extension property has right uniform deductive interpolation if and only if it is coherent and admits the amalgamation property.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Left Uniform Deductive Interpolation

V has left uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Π(y, z) | =V Σ(x, y) ⇐ ⇒ Π(y, z) | =V ∆(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Left Uniform Deductive Interpolation

V has left uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Π(y, z) | =V Σ(x, y) ⇐ ⇒ Π(y, z) | =V ∆(y).

Lemma

The following are equivalent: (1) V has left uniform deductive interpolation. (2) V has deductive interpolation, and for finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) has a left adjoint.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Left Uniform Deductive Interpolation

V has left uniform deductive interpolation if for any finite set of equations Σ(x, y), there exists a finite set of equations ∆(y) such that Π(y, z) | =V Σ(x, y) ⇐ ⇒ Π(y, z) | =V ∆(y).

Lemma

The following are equivalent: (1) V has left uniform deductive interpolation. (2) V has deductive interpolation, and for finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) has a left adjoint. Moreover, if V is locally finite, these are equivalent to (3) V has deductive interpolation, is congruence distributive, and for finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) preserves meets.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Model Completions

A first-order theory T ∗ is a model completion of a universal theory T if

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

Model Completions

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

Model Completions

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

Model Completions

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination. Moreover, T ∗ is then the theory of the existentially closed models for T.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

Model Completions

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination. Moreover, T ∗ is then the theory of the existentially closed models for T.

Theorem (Wheeler 1976)

The theory of V has a model completion if and only if V is coherent, admits the amalgamation property, and has the conservative congruence extension property for its finitely presented members.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

A General Theorem

Theorem (Ghilardi and Zawadowski 2002, van Gool et al. 2017)

• S. van Gool, G. Metcalfe, and C. Tsinakis.

Uniform interpolation and compact congruences. Annals of Pure and Applied Logic 168 (2017), 1927–1948.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

A General Theorem

Theorem (Ghilardi and Zawadowski 2002, van Gool et al. 2017)

Suppose that (i) V is coherent and has the amalgamation property;

• S. van Gool, G. Metcalfe, and C. Tsinakis.

Uniform interpolation and compact congruences. Annals of Pure and Applied Logic 168 (2017), 1927–1948.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

A General Theorem

Theorem (Ghilardi and Zawadowski 2002, van Gool et al. 2017)

Suppose that (i) V is coherent and has the amalgamation property; (ii) For finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) has a left adjoint, and KCon F(x) is dually Brouwerian.

• S. van Gool, G. Metcalfe, and C. Tsinakis.

Uniform interpolation and compact congruences. Annals of Pure and Applied Logic 168 (2017), 1927–1948.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

A General Theorem

Theorem (Ghilardi and Zawadowski 2002, van Gool et al. 2017)

Suppose that (i) V is coherent and has the amalgamation property; (ii) For finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) has a left adjoint, and KCon F(x) is dually Brouwerian. Then the theory of V has a model completion.

• S. van Gool, G. Metcalfe, and C. Tsinakis.

Uniform interpolation and compact congruences. Annals of Pure and Applied Logic 168 (2017), 1927–1948.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

A General Theorem

Theorem (Ghilardi and Zawadowski 2002, van Gool et al. 2017)

Suppose that (i) V is coherent and has the amalgamation property; (ii) For finite sets x, y, the compact lifting of F(y) ֒ → F(x, y) has a left adjoint, and KCon F(x) is dually Brouwerian. Then the theory of V has a model completion.

Corollary (Ghilardi and Zawadowski 1997)

The theory of Heyting algebras has a model completion.

• S. van Gool, G. Metcalfe, and C. Tsinakis.

Uniform interpolation and compact congruences. Annals of Pure and Applied Logic 168 (2017), 1927–1948.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

Tomorrow

We investigate uniform interpolation for some particular case studies, including varieties of modal algebras, lattices, and residuated lattices.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30

Tomorrow

We investigate uniform interpolation for some particular case studies, including varieties of modal algebras, lattices, and residuated lattices. We provide a general criterion for establishing the failure of coherence and hence also of uniform interpolation.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30

Tomorrow

We investigate uniform interpolation for some particular case studies, including varieties of modal algebras, lattices, and residuated lattices. We provide a general criterion for establishing the failure of coherence and hence also of uniform interpolation. We pose some open problems and challenges. . .

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30