A complete deductive calculus for (implications of) coequations - PowerPoint PPT Presentation

S -injectives Theorem. If U : C Γ � C has a right adjoint H and C has enough S -injectives, then C Γ has enough U − 1 S -injectives. Proof. Let � A, α � be given and A ≤ C , where C is S -injective. Then � A, α � ≤ HC . It suffices to show HC is U − 1 S -injective. A complete deductive calculus for (implications of) coequations – p.7/30

� � � S -injectives Theorem. If U : C Γ � C has a right adjoint H and C has enough S -injectives, then C Γ has enough U − 1 S -injectives. Proof. Let � A, α � be given and A ≤ C , where C is S -injective. Then � A, α � ≤ HC . It suffices to show HC is U − 1 S -injective. Let j : � B, β � � � � D, δ � and f : � B, β � � HC be given. � D, δ � HC � � � � j � � � f � � � B, β � A complete deductive calculus for (implications of) coequations – p.7/30

� � � � � � � S -injectives Theorem. If U : C Γ � C has a right adjoint H and C has enough S -injectives, then C Γ has enough U − 1 S -injectives. Proof. Let � A, α � be given and A ≤ C , where C is S -injective. Then � A, α � ≤ HC . It suffices to show HC is U − 1 S -injective. Let j : � B, β � � � � D, δ � and f : � B, β � � HC be given. � D, δ � HC D C � � � � � � � j � j � � � � f � � f � � � B, β � B A complete deductive calculus for (implications of) coequations – p.7/30

� � � � � � � S -injectives Theorem. If U : C Γ � C has a right adjoint H and C has enough S -injectives, then C Γ has enough U − 1 S -injectives. Proof. Let � A, α � be given and A ≤ C , where C is S -injective. Then � A, α � ≤ HC . It suffices to show HC is U − 1 S -injective. Let j : � B, β � � � � D, δ � and f : � B, β � � HC be given. By the injectivity of C , we get a map D � C as shown . . . � D, δ � � C HC D � � � � � � � j � j � � � � f f � � � � � B, β � B A complete deductive calculus for (implications of) coequations – p.7/30

� � � � � � � S -injectives Theorem. If U : C Γ � C has a right adjoint H and C has enough S -injectives, then C Γ has enough U − 1 S -injectives. Proof. Let � A, α � be given and A ≤ C , where C is S -injective. Then � A, α � ≤ HC . It suffices to show HC is U − 1 S -injective. Let j : � B, β � � � � D, δ � and f : � B, β � � HC be given. By the injectivity of C , we get a map D � C as shown and hence, by adjoint transposition, a homomorphism � D, δ � � HC . � HC � D, δ � � C D � � � � � � � j � j � � � � f f � � � � � B, β � B A complete deductive calculus for (implications of) coequations – p.7/30

� � � � � About S -meets Recall that Sub ( C ) denotes the poset of isomorphism classes of S -morphisms into C . In any factorization system �H , S� , the S -morphisms are stable under pullbacks. h ∗ A A �� � C B h Thus, if C has pullbacks of S -morphisms, then each � Sub ( B ) . � C induces a functor h ∗ : Sub ( C ) h : B A complete deductive calculus for (implications of) coequations – p.8/30

� � � � � � � About S -meets In any factorization system �H , S� , the S -morphisms are This gives one a notion of ∧ for stable under pullbacks. � Sub ( C ) . Sub ( C ) , ∧ : Sub ( C ) × Sub ( C ) A ∧ B A �� � C B h A complete deductive calculus for (implications of) coequations – p.8/30

� � � � � � � � About S -meets In any factorization system �H , S� , the S -morphisms are stable under generalized pullbacks. A � � � ���������������� � � � �������� � � � � � � � � � � � � � . . . . . . A i 0 A i 1 � A i 2 A i κ � � � � ���������������� � � � � � � � � � � � � � � � � � � � � � C A complete deductive calculus for (implications of) coequations – p.8/30

� � � � � � � � � � � � About S -meets In any factorization system �H , S� , the S -morphisms are Assuming that C stable under generalized pullbacks. � I for Sub ( C ) , has such limits, this gives one a notion of � � Sub ( C ) . I : Sub ( C ) I � I A i � � ��������������� � � � � � � ������� � � � � � � � � � � . . . . . . A i 0 A i 1 A i 2 A i κ � � � ������������������ � � � � � � � � � � � � � � � � � � � � � � � � � C A complete deductive calculus for (implications of) coequations – p.8/30

Structural summary • If C has coproducts, then so does C Γ . A complete deductive calculus for (implications of) coequations – p.9/30

Structural summary • If C has coproducts, then so does C Γ . • If C has a factorization system �H , S� and Γ preserves S -morphisms, then C Γ has a factorization system � U − 1 H , U − 1 S� . A complete deductive calculus for (implications of) coequations – p.9/30

Structural summary • If C has coproducts, then so does C Γ . • If C has a factorization system �H , S� and Γ preserves S -morphisms, then C Γ has a factorization system � U − 1 H , U − 1 S� . • If C is S -well-powered, then C Γ is U − 1 S -well-powered. A complete deductive calculus for (implications of) coequations – p.9/30

Structural summary • If C has coproducts, then so does C Γ . • If C has a factorization system �H , S� and Γ preserves S -morphisms, then C Γ has a factorization system � U − 1 H , U − 1 S� . • If C is S -well-powered, then C Γ is U − 1 S -well-powered. • If C has enough S -injectives and U : C Γ � C has a right adjoint, then C Γ has enough U − 1 S -injectives. A complete deductive calculus for (implications of) coequations – p.9/30

Structural summary Hereafter, we assume that C has all coproducts, a factor- ization system �H , S� , enough S -injectives and meets of S -morphisms and is S -well-powered, and that Γ -preserves S -morphisms. We further assume that U : C Γ � C has a right adjoint H . A complete deductive calculus for (implications of) coequations – p.9/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.10/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.10/30

Quasi-covarieties and covarieties Let V ⊆ C Γ . We define H V = {� B, β � | ∃ V ∋ � C, γ � � � � B, β �} S V = {� B, β � | ∃� B, β � � � � C, γ � ∈ V } � Σ V = { � C i , γ i � | � C i , γ i � ∈ V } A complete deductive calculus for (implications of) coequations – p.11/30

Quasi-covarieties and covarieties Let V ⊆ C Γ . We define H V = {� B, β � | ∃ V ∋ � C, γ � � � � B, β �} S V = {� B, β � | ∃� B, β � � � � C, γ � ∈ V } � Σ V = { � C i , γ i � | � C i , γ i � ∈ V } We say that V is a quasi-covariety if V ⊆ H Σ V . A complete deductive calculus for (implications of) coequations – p.11/30

Quasi-covarieties and covarieties Let V ⊆ C Γ . We define H V = {� B, β � | ∃ V ∋ � C, γ � � � � B, β �} S V = {� B, β � | ∃� B, β � � � � C, γ � ∈ V } � Σ V = { � C i , γ i � | � C i , γ i � ∈ V } We say that V is a quasi-covariety if V ⊆ H Σ V . We say that V is a covariety if V ⊆ SH � V . A complete deductive calculus for (implications of) coequations – p.11/30

A coequational language Fix a S -injective C ∈ C . We define a simple language L Coeq (properly, L C Coeq ). • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . A complete deductive calculus for (implications of) coequations – p.12/30

✁ A coequational language Fix a S -injective C ∈ C . We define a simple language L Coeq (properly, L C Coeq ). • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . A complete deductive calculus for (implications of) coequations – p.12/30

✁ A coequational language Fix a S -injective C ∈ C . We define a simple language L Coeq (properly, L C Coeq ). • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . A complete deductive calculus for (implications of) coequations – p.12/30

✁ A coequational language Fix a S -injective C ∈ C . We define a simple language L Coeq (properly, L C Coeq ). • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . A complete deductive calculus for (implications of) coequations – p.12/30

✁ A coequational language Fix a S -injective C ∈ C . We define a simple language L Coeq (properly, L C Coeq ). • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . A complete deductive calculus for (implications of) coequations – p.12/30

☎ ✂ ☎ ✄ ✄ A coequational language • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . � Sub ( UHC ) : − : L Coeq We define an interpretation = P P A complete deductive calculus for (implications of) coequations – p.12/30

☎ ✂ ✄ ✂ ✂ ✄ ☎ ☎ ✄ ✂ A coequational language • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . � Sub ( UHC ) : − : L Coeq We define an interpretation = ϕ ϕ (Definition of forthcoming!) A complete deductive calculus for (implications of) coequations – p.12/30

✄ ☎ ☎ ✄ ✂ ☎ ✄ A coequational language • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . � Sub ( UHC ) : − : L Coeq We define an interpretation � � = ϕ i ϕ i A complete deductive calculus for (implications of) coequations – p.12/30

☎ ✄ ☎ ✄ ✂ ☎ ✄ A coequational language • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . � Sub ( UHC ) : − : L Coeq We define an interpretation = h ∗ ϕ ( h ( x )) ϕ A complete deductive calculus for (implications of) coequations – p.12/30

☎ ✄ ☎ ✄ ✂ ☎ ✄ A coequational language • For every P in Sub ( UHC ) , we introduce an atomic proposition P in L Coeq , i.e., Sub ( UHC ) ⊆ L Coeq . • If ϕ ∈ L Coeq , then ϕ ∈ L Coeq . � • If { ϕ i } i ∈ I ⊆ L Coeq , then I ϕ i ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ϕ ( h ( x )) ∈ L Coeq . • If ϕ ∈ L Coeq and h : HC � HC , then ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) is in L Coeq . � Sub ( UHC ) : − : L Coeq We define an interpretation ∃ y ( ϕ ( y ) ∧ h ( y ) = x ) = ∃ h ϕ A complete deductive calculus for (implications of) coequations – p.12/30

✝ ✆ Coequations A coalgebra � A, α � satisfies ϕ iff for every homomorphism p : � A, α � � HC , we have Im ( p ) ≤ ϕ . A complete deductive calculus for (implications of) coequations – p.13/30

� ✆ � � ✆ ✝ ✝ ✆ � ✝ Coequations A coalgebra � A, α � satisfies ϕ iff for every homomorphism p : � A, α � � HC , we have Im ( p ) ≤ ϕ . In other words, � A, α � | = ϕ iff every p : � A, α � � HC factors through ϕ . p A UHC � � � � � � � � � ϕ A complete deductive calculus for (implications of) coequations – p.13/30

� ✆ ✝ ✆ � ✝ � � Coequations � A, α � | = ϕ iff every p : � A, α � � HC factors through ϕ . p A UHC � � � � � � � � � ϕ Homomorphisms p : � A, α � � HC correspond to colorings p : A � C . Thus, � A, α � | = ϕ just in case, however we color � A (via � p ), the image of the corresponding homomorphism p lies in ϕ . A complete deductive calculus for (implications of) coequations – p.13/30

Example The cofree coalgebra H 2 A complete deductive calculus for (implications of) coequations – p.14/30

Example A coequation. A complete deductive calculus for (implications of) coequations – p.14/30

Example This coalgebra satisfies P . A complete deductive calculus for (implications of) coequations – p.14/30

Example Under any coloring, the elements of the coalgebra map to elements of P . A complete deductive calculus for (implications of) coequations – p.14/30

Example This coalgebra doesn’t satisfy P . A complete deductive calculus for (implications of) coequations – p.14/30

Example If we paint the circle red, it isn’t mapped to an element of P . A complete deductive calculus for (implications of) coequations – p.14/30

✝ ✆ ✆ ✝ An implicational language Define L Imp = { ϕ ⇒ ψ | ϕ, ψ ∈ L Coeq } . Say that � A, α � | = ϕ ⇒ ψ just in case, for every p : � A, α � � HC such that Im ( p ) ≤ ϕ , also Im ( p ) ≤ ψ . A complete deductive calculus for (implications of) coequations – p.15/30

� ✝ � � � ✆ ✆ ✝ ✝ ✆ ✆ � ✝ � An implicational language Define L Imp = { ϕ ⇒ ψ | ϕ, ψ ∈ L Coeq } . Say that � A, α � | = ϕ ⇒ ψ just in case, for every p : � A, α � � HC such that Im ( p ) ≤ ϕ , also Im ( p ) ≤ ψ . p � p � A UHC A UHC � � � � � � � � � � ⇒ � � � � � � � � � Q P This is not the same as ( � A, α � �| = ϕ or � A, α � | = ψ ). That would be true if either there is some p such that Im ( p ) �≤ ϕ or for all p , Im ( p ) ≤ ψ . A complete deductive calculus for (implications of) coequations – p.15/30

� ✆ � � � � � ✝ ✝ ✆ An implicational language Define L Imp = { ϕ ⇒ ψ | ϕ, ψ ∈ L Coeq } . Say that � A, α � | = ϕ ⇒ ψ just in case, for every p : � A, α � � HC such that Im ( p ) ≤ ϕ , also Im ( p ) ≤ ψ . p � p � A UHC A UHC � � � � � � � � � � ⇒ � � � � � � � � � Q P This is also not the same as � A, α � | = ¬ ϕ ∨ ψ (if Sub ( UHC ) is a Heyting algebra). A complete deductive calculus for (implications of) coequations – p.15/30

� ✆ � � � � ✆ � ✝ ✝ An implicational language Define L Imp = { ϕ ⇒ ψ | ϕ, ψ ∈ L Coeq } . Say that � A, α � | = ϕ ⇒ ψ just in case, for every p : � A, α � � HC such that Im ( p ) ≤ ϕ , also Im ( p ) ≤ ψ . p � p � A UHC A UHC � � � � � � � � � � ⇒ � � � � � � � � � Q P Note: � A, α � | = ϕ iff � A, α � | = ⊤ ⇒ ϕ, where ⊤ = ( HC HC ) . A complete deductive calculus for (implications of) coequations – p.15/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.16/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.16/30

The Covariety Theorems Given a class V of coalgebras, define Th ( V ) = { ϕ ∈ L C Coeq | V | = ϕ, C S -injective } , Imp ( V ) = { P ⇒ Q ∈ L C Imp | V | = P ⇒ Q, P, Q ≤ UHC, C S -injective } . A complete deductive calculus for (implications of) coequations – p.17/30

The Covariety Theorems Given a class V of coalgebras, define Th ( V ) = { ϕ ∈ L C Coeq | V | = ϕ, C S -injective } , Imp ( V ) = { P ⇒ Q ∈ L C Imp | V | = P ⇒ Q, P, Q ≤ UHC, C S -injective } . Given a collection S of (implications between) coequations, define Mod ( S ) = {� A, α � | � A, α � | = S } . A complete deductive calculus for (implications of) coequations – p.17/30

The Covariety Theorems Theorem (The “co-Birkhoff” theorem). For any V , SH Σ V = Mod Th ( V ) . A complete deductive calculus for (implications of) coequations – p.17/30

The Covariety Theorems Theorem (The “co-Birkhoff” theorem). For any V , SH Σ V = Mod Th ( V ) . Theorem (The co-quasivariety theorem). For any V , H Σ V = Mod Imp ( V ) . A complete deductive calculus for (implications of) coequations – p.17/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.18/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.18/30

Birkhoff’s completeness theorem So much for the formal dual of the variety theorem. What about the formal dual of Birkhoff’s completeness theorem? A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem So much for the formal dual of the variety theorem. What about the formal dual of Birkhoff’s completeness theorem? Let S be a set of equations for an algebraic signature Σ . Let Ded ( S ) denote the deductive closure of S under the usual equational logic. A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem So much for the formal dual of the variety theorem. What about the formal dual of Birkhoff’s completeness theorem? Let S be a set of equations for an algebraic signature Σ . Let Ded ( S ) denote the deductive closure of S under the usual equational logic. Theorem (Birkhoff’s completeness theorem). For any set S of equations, Ded ( S ) = Th Mod ( S ) . A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem So much for the formal dual of the variety theorem. What about the formal dual of Birkhoff’s completeness theorem? Let S be a set of equations for an algebraic signature Σ . Let Ded ( S ) denote the deductive closure of S under the usual equational logic. Theorem (Birkhoff’s completeness theorem). For any set S of equations, Ded ( S ) = Th Mod ( S ) . Here, Th ( V ) denotes the equational theory of a class of al- gebras. A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem Theorem (Birkhoff’s completeness theorem). For any set S of equations, Ded ( S ) = Th Mod ( S ) . Compare this to the variety theorem, namely for every V , HSP V = Mod Th ( V ) . A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem Theorem (Birkhoff’s completeness theorem). For any set S of equations, Ded ( S ) = Th Mod ( S ) . Main goal Find a logic on sets of coequations such that for any set S of coequations over C , Ded ( S ) = Th Mod ( S ) . A complete deductive calculus for (implications of) coequations – p.19/30

Birkhoff’s completeness theorem Theorem (Birkhoff’s completeness theorem). For any set S of equations, Ded ( S ) = Th Mod ( S ) . Main goal Find a logic on sets of coequations such that for any set S of coequations over C , Ded ( S ) = Th Mod ( S ) . First step Find the formal dual to Birkhoff’s completeness theorem. A complete deductive calculus for (implications of) coequations – p.19/30

✁ ✞ ✁ ✞ The invariance theorem Define interior operators � Sub ( UHC ) , : Sub ( UHC ) by � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✁ ✁ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } • P is the (carrier of the) largest subcoalgebra of HC . A complete deductive calculus for (implications of) coequations – p.20/30

✁ ✞ ✞ ✞ ✞ ✁ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } • P is the (carrier of the) largest subcoalgebra of HC . • P is the largest endomorphism invariant subobject of UHC , that is: • For every h : HC � HC , ∃ h P ≤ P ; A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✞ ✞ ✁ ✁ ✞ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } • P is the (carrier of the) largest subcoalgebra of HC . • P is the largest endomorphism invariant subobject of UHC , that is: • For every h : HC � HC , ∃ h P ≤ P ; • If, for every h : HC � HC , ∃ h Q ≤ Q , then Q ≤ P . A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✞ ✞ ✞ ✞ ✞ ✞ ✁ ✞ ✞ ✞ ✞ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } is an S4 necessity operator. • If P ⊢ Q then P ⊢ Q ; • P ⊢ P ; • P ⊢ P ; • ( P → Q ) ⊢ P → Q ; A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✞ ✞ ✞ ✞ ✞ ✞ ✁ ✞ ✁ ✞ ✞ ✞ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } is an S4 necessity operator. • If P ⊢ Q then P ⊢ Q ; • P ⊢ P ; • P ⊢ P ; • ( P → Q ) ⊢ P → Q ; If Γ preserves pullbacks of S -morphisms, then so is . A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✁ ✁ ✞ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } Theorem (The invariance theorem). Let ϕ be a coequation over C . For any coequation ψ over C , Mod ( ϕ ) | = ψ iff ϕ ≤ ψ . A complete deductive calculus for (implications of) coequations – p.20/30

✞ ✞ ✁ ✞ ✁ ✁ The invariance theorem � P = { U � A, α � � � UHC | � A, α � ∈ Sub C Γ ( HC ) } � P = { Q � � UHC | ∀ h : HC � HC . ∃ h Q ≤ P } Theorem (The invariance theorem). Let ϕ be a coequation over C . For any coequation ψ over C , Mod ( ϕ ) | = ψ iff ϕ ≤ ψ . P is the least coequation satisfied by In other words, Mod ( P ) . It can be regarded as a measure of the “coequa- tional commitment” of P . A complete deductive calculus for (implications of) coequations – p.20/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.21/30

Outline I. Preliminaries II. Quasi-covarieties and covarieties III. Coequations IV. The Covariety Theorems V. The Invariance Theorem VI. Coequational logic (Soundness) VII. Coequational logic (Completeness) VIII. Implicational logic (Soundness) IX. Implicational logic (Completeness) A complete deductive calculus for (implications of) coequations – p.21/30

A sound rule An inference rule ϕ 1 . . . ϕ n is sound just in case, ψ whenever � A, α � | = ϕ 1 , ..., � A, α � | = ϕ n , then � A, α � | = ψ . A complete deductive calculus for (implications of) coequations – p.22/30

A sound rule An inference rule ϕ 1 . . . ϕ n is sound just in case, ψ whenever � A, α � | = ϕ 1 , ..., � A, α � | = ϕ n , then � A, α � | = ψ . � ϕ i � Theorem. The rule is sound. -E ϕ i A complete deductive calculus for (implications of) coequations – p.22/30