General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Contents 1. General Problem 2. Quasi-primal algebras Logics associated with a quasi-primal algebra Protoalgebraic logics Truth-equational logics Algebraizable logics Tommaso Moraschini An example Joint work with Prof. Josep Maria Font 3. Primal algebras Logics of g-matrices June 26, 2014 Protoalgebraic logics Algebraizable logics An example 4. Ubiquitous algebraizability 1 / 32 2 / 32 General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Abstract Algebraic Logic Abstract Algebraic Logic Fix a logic L . Two things may happen: ◮ A logic L is a substitution invariant closure operator ◮ The Leibniz congruence admits a nice description. C L : P ( Fm ) → P ( Fm ) . L is protoalgebraic if there is a set of formulas ∆( x , y , z ) s.t. for every algebra A , F ∈ F i L A and a , b ∈ A : ◮ Pick an algebra A . The subsets of A closed under the rules of L are the filters F i L A of L over A . � a , b � ∈ Ω A F ⇐ ⇒ ∆ A ( a , b , c ) ⊆ F for every c ∈ A . ◮ Pick any F ⊆ A . The Leibniz congruence Ω A F is the greatest congruence on A compatible with F . For IPC pick ∆( x , y , z ) = { x → y , y → x } . ◮ The class of reduced models of L is ◮ Truth predicates in Mod ∗ L have a nice description. L is truth-equational if there is a set of equations τ ( x ) s.t. Mod ∗ L = {� A , F � : F ∈ F i L A and Ω A F = Id A } . F = { a ∈ A : A | = τ ( a ) } L is complete w.r.t. Mod ∗ L . for every � A , F � ∈ Mod ∗ L . Example: Mod ∗ IPC = {� A , { 1 }� : A is an Heyting algebra } . For IPC pick τ ( x ) = { x ≈ 1 } . 4 / 32 5 / 32
� � � � � � � � � � General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Abstract Algebraic Logic General problem regularly algebraizable ◮ What can we say about the logic of a (finite) matrix � A , F � ? ◮ Can we classify it within the Leibniz hierarchy somehow? regularly weakly Yes, mechanically. algebraizable algebraizable ◮ Can we classify it within the Leibniz hierarchy in a nicer way? Yes, for A being a quasi-primal algebra. weakly ◮ How do algebraizable logics of a variety V look like? Are they equivalential assertional algebraizable determined by a finite matrix? For varieties generated by a (finite) quasi-primal algebra. protoalgebraic truth-equational Figure : Some classes of the Leibniz hierarchy. 6 / 32 7 / 32 General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Quasi-primal algebras Protoalgebraic logics When is a logic of � A , F � protoalgebraic? Given a set A , the ternary discriminator function on A is the map Lemma t : A 3 → A such that Let A be a quasi-primal algebra and C a non-almost inconsistent � a if a � = b closure system over A . The logic determined by � A , C� is t ( a , b , c ) = protoalgebraic if and only if it has theorems. c otherwise for every a , b , c ∈ A . Proof. ◮ Pick a theorem ϕ ( x ) at most in variable x . Definition ◮ Check that An algebra A is quasi-primal if there is a term t ( x , y , z ) which � � ∆( x , y ) := { t y , x , ϕ ( x ) } . represents the ternary discriminator term on A , i.e., such that t A ( x , y , z ) is the ternary discriminator function of A . is a set of protoimplication formulas for L . 9 / 32 10 / 32
General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Truth-equational logics A counterexample A counterexample to direction (i) ⇒ (ii). Let A = �{ a , b , ⊤} , ✷ , ✸ , ⊤� be the algebra with unary-operations ✷ When is a logic of � A , F � truth-equational? and ✸ defined as Theorem ✷ a = ✷ b = b ✷ ⊤ = ⊤ Let A be a quasi-primal algebra, τ ( x ) a set of equations, F ∈ P ( A ) � { A } and L the logic determined by � A , F � . The ✸ b = ✸ ⊤ = ⊤ ✸ a = b . following conditions are equivalent: Let L be the logic of � A , { a , ⊤}� . We have that: (i) τ ( x ) defines truth in � A , F � and L has theorems. ◮ L has theorems. (ii) L is truth-equational via τ ( x ) . ◮ { a , ⊤} is equationally definable by { ✷ x ≈ ✸ x } . (iii) L is weakly-algebraizable via τ ( x ) . ◮ � A , { a , ⊤}� ∈ Mod ∗ L . The equivalence of (i) and (ii) is not true in general! It is possible to prove that also � A , {⊤}� ∈ Mod ∗ L . Hence truth is not not implicitly definable in Mod ∗ L . 11 / 32 12 / 32 General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Truth-equational logics Algebraizable logics Corollary Theorem Let A be quasi-primal. The following conditions are equivalent: Let A be a non-trivial finite quasi-primal algebra. The following (i) There is a closure system C ⊆ P ( A ) s.t. � A , C� determines a conditions are equivalent: non-trivial protoalgebraic logic. (ii) There is an unary term-function ¬ A : A → A that is not (i) L is algebraizable with equivalent algebraic semantics V ( A ) . surjective. (ii) L has theorems and is the logic determined by � A , F � , for some F ⊆ A such that F is equationally definable and For A finite we can strengthen (i) as: B ∩ F � = B for every non-trivial B ∈ S ( A ) . (i’) There is a closure system C ⊆ P ( A ) s.t. � A , C� determines a non-trivial weakly-algebraizable logic. 13 / 32 14 / 32
� � � � � General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Algebraizable logics Logics of quasi-primal algebras Some sufficient and necessary conditions (or a normal form for Summary. For the logic of a g-matrix � A , C� with C ⊆ A non-trivial algebraizable logics of finite quasi-primal algebras, up to deductive closure system and A non-trivial, finite and quasi-primal we have: equivalence): protoalgebraic ← → having theorems Corollary truth-equational ← → having theorems + C = { F , A } Let A finite, non-trivial and quasi-primal. TFAE: for some F equationally definable (i) There is an algebraizable logic of V ( A ) . algebraizable ← → truth-equational + F ∩ B � = B (ii) There is an algebraizable logic of V ( A ) with ρ ( x , y ) = { x ↔ y } and τ ( x ) = { x ↔ x ≈ x } for some term for every non-trivial B ∈ S ( A ) . x ↔ y . Moreover: (iii) There is a term x ↔ y s.t. x ↔ x : A → A is idempotent and ◮ truth-equational ← → weakly-algebraizable. non-surjective and for every a , b ∈ A ◮ Every algebraizable logic of V ( A ) is of the kind above. a ↔ b ∈ { c ↔ c : c ∈ A } = ⇒ a = b . 15 / 32 16 / 32 General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability General Problem Quasi-primal algebras Primal algebras Ubiquitous algebraizability Logics of quasi-primal algebras An example There are 6 matrices on Ł Ł 3 , which do not determine a trivial logic. Ł regularly The truth predicate of each of them is equationally definable as algebraizable follows: � Ł Ł Ł 3 , { 1 }� �− → { x ≈ 1 } algebraizable Ł 3 , { 1 � Ł Ł 2 }� �− → { x ⊕ x ≈ 1 , x ⊙ x ≈ 0 } � Ł Ł Ł 3 , { 0 }� �− → { x ≈ 0 } weakly equivalential Ł 3 , { 1 algebraizable � Ł Ł 2 , 1 }� �− → { x ⊕ x ≈ 1 } Ł 3 , { 0 , 1 � Ł Ł 2 }� �− → { x ⊙ x ≈ 0 } protoalgebraic � Ł Ł Ł 3 , { 0 , 1 }� �− → { x ⊕ x ≈ x } . Figure : Logics of finite quasi-primal algebras. 17 / 32 18 / 32
Recommend
More recommend
