Many-Sorted First-Order Model Theory Lecture 2 12 th June, 2020 1 / - PowerPoint PPT Presentation

Many-Sorted First-Order Model Theory Lecture 2 12 th June, 2020 1 / 22

Conservative signature morphisms Definition 1 (Conservative signature morphism) A signature morphism χ : Σ → Σ ′ is conservative iff each Σ-model has a χ -expansion. Exercise 1 ◮ A signature morphism χ : Σ → Σ ′ is conservative iff it is injective. ◮ For any conservative signature morphism χ : Σ → Σ and any sets of Σ -sentences Γ 1 and Γ 2 we have Γ 1 | = Σ Γ 2 iff χ (Γ 1 ) | = Σ ′ χ (Γ 2 ) . ◮ Show that χ (Γ 1 ) | = Σ ′ χ (Γ 2 ) implies Γ 1 | = Σ Γ 2 doesn’t hold if χ is not conservative. 2 / 22

Term models Definition 2 (Non-void signatures) A signature Σ is non-void if all sorts are inhabited by some terms, i.e. T Σ , s � = ∅ for all s ∈ S . Fact 3 (Term models) If Σ is non-void then the set of Σ -terms T Σ can be regarded as a Σ -model: ◮ the carrier set for each sort s ∈ S is T Σ , s (which is not empty); ◮ for each σ : s 1 . . . s n → s ∈ F, the function σ T Σ : T Σ , s 1 × · · · × T Σ , s n → T Σ , s is defined by σ T Σ ( t 1 , . . . , t n ) = σ ( t 1 , . . . , t n ) for all i ∈ { 1 , . . . , n } and t i ∈ T Σ , s i ; ◮ T Σ interprets each relation symbol as the empty set, i.e. π T Σ = ∅ for all ( π : w ) ∈ P. 3 / 22

More on term models Theorem 4 (Initiality) Given any non-void signature Σ and any Σ -model A , there is a unique homomorphism h : T Σ → A defined by the interpretation of each term into the model A , i.e. h ( t ) = t A for all sorts s ∈ S and terms t ∈ T Σ , s . Proof. Assume there exists another homomorphism g : T Σ → A . We prove that g ( t ) = h ( t ) for all sorts s ∈ S and terms t ∈ T Σ , s . We proceed by induction on the structure of terms: ◮ h ( σ ( t 1 , . . . , t n )) = σ A ( h ( t 1 ) , . . . , h ( t n )) IH = σ A ( g ( t 1 ) , . . . , g ( t n )) = g ( σ ( t 1 , . . . , t n )). 4 / 22

� � � � � � � � More on term models Notation Σ[ X ] T Σ[ X ] Let Σ be a signature and X a set of vari- ables for Σ such that Σ[ X ] is non-void. ι X ↾ ι X We let T Σ ( X ) denote T Σ[ X ] ↾ Σ , the free Σ T Σ ( X ) = T Σ[ X ] ↾ Σ model of terms with variables from X . Exercise 2 (Freeness) Assume v # � A ◮ a signature Σ and a set of variables X T Σ ( X ) for Σ such that Σ[ X ] is non-void, and u v ◮ a Σ -model A and an evaluation X v : X → | A | . Then there exists a unique homomorphism v # : T Σ ( X ) → A such that v = u ; v # , where T Σ ( X ) = T Σ[ X ] ↾ Σ and u : X ֒ → T Σ ( X ) . 5 / 22

Substructures Definition 5 (Substructure/submodel) Let Σ be a signature and A and B two Σ-models. A is a substructure of B , in symbols A ⊆ B , iff there exists an inclusion homomorphism h : A → B . Exercise 3 1. | A | ⊆ | B | , A ⊆ B iff 2. σ B ( a ) ∈ A s for all ( σ : w → s ) ∈ F and a ∈ A w , 3. π A ⊆ π B . Exercise 4 Let Σ be a signature and h : A → B a Σ -homomorphism. Then h ( A ) is a submodel of B . Note that π h ( A ) = { h ( a ) | a ∈ π A } . 6 / 22

Congruence Definition 6 (Congruence) Let Σ be a signature and A be a Σ-model. A congruence ≡ = {≡ s } s ∈ S on A is 1. an equivalence on | A | , i.e. an S -sorted relation ≡ s ⊆ A s × A s for all s ∈ S satisfying the following properties: ◮ ( Reflexivity ) a ≡ s a for all s ∈ S and a ∈ A s ◮ ( Symmetry ) a 1 ≡ s a 2 for all s ∈ S and a 1 , a 2 ∈ A s a 2 ≡ s a 1 ◮ ( Transitivity ) a 1 ≡ s a 2 a 2 ≡ s a 3 for all s ∈ S and a 1 , a 2 , a 3 ∈ A s a 1 ≡ s a 3 2. compatible with the function symbols in Σ a 1 ≡ s 1 a ′ 1 . . . a n ≡ s n a ′ n ◮ ( Congruence ) σ A ( a 1 , . . . , a n ) ≡ s σ A ( a ′ 1 , . . . , a ′ n ) for all ( σ : s 1 . . . s n → s ) ∈ F and a i , a ′ i ∈ A s i for all i ∈ { 1 , . . . , n } . We will drop the subscript s from ≡ s whenever it is clear from the context. Convention Let ≡ be a congruence on a model A . If (a) w = s 1 . . . s n ∈ S ∗ (b) a = a 1 . . . a n ∈ A w , and (c) a ′ = a ′ n ∈ A w then a ≡ w a ′ iff a i ≡ s i a ′ 1 . . . a ′ i for all i ∈ { 1 , . . . , n } . 7 / 22

Examples of congruences Example 8 (INT) Example 7 (NAT) spec INT is sort Int . spec NAT is op 0 : -> Int . sort Nat . op s_ : Int -> Int . op 0 : -> Nat . op p_ : Int -> Int . op s_ : Nat -> Nat . end end Let ≡ be the congruence on T (Σ INT ) We define the congruence ≡ on N as follows: generated by the following two sets of pairs of terms: ◮ n 1 ≡ n 2 iff ( n 1 mod 2) = ( n 2 mod 2) ◮ { p s t ≡ t | t ∈ T (Σ INT ) } , and for all n 1 , n 2 ∈ N . ◮ { s p t ≡ t | t ∈ T (Σ INT ) } . Exercise 5 Exercise 6 Prove that ≡ defined on N above is a Prove that the intersection of two congruence. congruences is again a congruence. Exercise 7 Let E be a set of equations over a non-void signature Σ . Prove that ≡ E := { ( t 1 , t 2 ) | E | = t 1 = t 2 } is the least congr. on T Σ generated by { ( t 1 , t 2 ) | ( t 1 = t 2 ) ∈ E } . 8 / 22

Kernel Lemma 9 Let Σ be a signature and h : A → B a Σ -homomorphism. We define the congruence ker( h ) = { ker( h ) s } s ∈ S on A as follows: ◮ ker( h ) s = { ( a , b ) | h s ( a ) = h s ( b ) } for all s ∈ S The relation ker( h ) is a congruence on A . Proof. ◮ ( Reflexivity ): Obviously, h ( a ) = h ( a ), which implies ( a , a ) ∈ ker( h ) ◮ ( Symmetry ): We assume ( a , b ) ∈ ker( h ) and we show that ( b , a ) ∈ ker( h ). We have: a ker( h ) b iff h ( a ) = h ( b ) iff h ( b ) = h ( a ) iff b ker( h ) a . ◮ ( Transitivity ): We assume ( a , b ) ker( h ) and ( b , c ) ∈ ker( h ), and we show that ( a , c ) ∈ ker( h ). Since ( a , b ) ∈ ker( h ) and ( b , c ) ∈ ker( h ), we have h ( a ) = h ( b ) and h ( b ) = h ( c ). We obtain h ( a ) = h ( c ). Hence, ( a , c ) ∈ ker( h ). ◮ ( Congruence ): Let ( σ : w → s ) ∈ F . We assume that ( a , b ) ∈ ker( h ) w and we prove that ( σ A ( a ) , σ A ( b )) ∈ ker( h ) s . Since ( a , b ) ∈ ker( h ) w , we have h w ( a ) = h w ( b ). It follows that h s ( σ A ( a )) = σ B ( h w ( a )) = σ B ( h w ( b )) = h s ( σ A ( b )). Hence, ( σ A ( a ) , σ A ( b )) ∈ ker( h ) s . 9 / 22

Quotient Notation Let ≡ be a congruence on a Σ-model A . Let s ∈ S and a ∈ A s . a := { a ′ ∈ A s | a ≡ a ′ } sometimes denoted also by a / ≡ . The class of a is � A s Fact 10 � a Note that � a ⊆ A s for all s ∈ S and a ∈ A s . This means that ≡ determines a partition of the universe | A | . Example 11 Example 12 Consider the congruence defined Consider the congruence defined in Example 8. in Example 7. We have: We have: ◮ � ◮ � 0 = { 0 , 2 , 4 , 6 , . . . } 0 = { 0 , p s 0 , s p 0 , p s p s 0 , . . . } ◮ � ◮ � 1 = { 1 , 3 , 5 , 7 , . . . } p 0 = { p 0 , p s p 0 , s p p 0 , p s p s p 0 . . . } 10 / 22

Convention If w = s 1 . . . s n and a = ( a 1 , . . . , a n ) ∈ A w then we let � a denote the tuple ( � a 1 , . . . , � a n ). Definition 13 (Quotient substructures) Let ≡ be a congruence on a Σ-model A . The quotient structure of A modulo ≡ is the Σ-structure � A (also denoted A / ≡ ) defined below: ◮ � A s = { � a | a ∈ A s } for all sorts s ∈ S , ◮ for all function symbols ( σ : w → s ) ∈ F , A : � a ) = � the function σ � A w → � A s is defined by σ � A ( � σ A ( a ) for all a ∈ A w ; ◮ for all relation symbols ( π : w ) ∈ P , A is defined by π � A = { � a | a ′ ∈ π A for some a ′ ∈ � the relation π � a } . Lemma 14 � A is well-defined. Proof. A : � σ � A w → � a = � b then a ≡ b , which implies σ A ( a ) ≡ σ A ( b ), and we get A s is a function: if � a ) = � σ A ( a ) = � σ � σ A ( b ) = σ � A ( � A ( � b ). 11 / 22

� � First isomorphism theorem Fact 15 The quotient map q : A → � A defined by q ( a ) = � a for all s ∈ S and a ∈ A s is a homomorphism. Theorem 16 (First isomorphism theorem) ∃ ! g � � B A Let h : A → B be a Σ -homomorphism and let ≡ be a � congruence on A such that ≡⊆ ker( h ) . There exists a q h unique homomorphism g : � A → B such that q ; g = h, A where q is the quotient homomorphism q : A → � A . Proof. We define g : � A → B by g ( � a ) = h ( a ) for all s ∈ S and a ∈ A s . We show that g is well-defined: a ′ then a ≡ a ′ ; since ≡⊆ ker( h ), h ( a ) = h ( a ′ ); ◮ if � a = � ◮ since h is compatible with the function and relation symbols, g is compatible with the function and relation symbols too. For the uniqueness part, let f : � A → B such that q ; f = q ; g . For all s ∈ S and a ∈ A s , f ( � a ) = f ( q ( a )) = g ( q ( a )) = g ( � a ). 12 / 22

� � Second isomorphism theorem/homomorphic image theorem ∃ ! g � � im ( h ) A Theorem 17 (Second isomorphism theorem) � Let h : A → B be a Σ -homomorphism. q h Then A / ker( h ) ∼ = im ( h ) . A Proof. By a slightly abuse of notation we denote by h the co-restriction of h : A → B to im ( h ). Let q : A → � A by the quotient map from Fact 15. We apply Theorem 16 with ≡ = ker( h ). There exists a unique homomorphism g : � A → im ( h ) such that q ; g = h . Clearly g is surjective. We need to show that g is injective: a ) = g ( � a ) = g ( � a ′ ) = h ( a ′ ). a ′ ). We have h ( a ) = g ( � Assume that g ( � a = � Since ≡ = ker( h ), we get � a ′ . 13 / 22