An algebraic approach to canonical formulas Nick Bezhanishvili Imperial College London Joint work with Guram Bezhanishvili New Mexico State University 1 / 41
Jankov formulas Refutation patterns play an important role in developing axiomatic bases for intermediate and modal logics. First algebra-based formulas for finite subdirectly irreducible Heyting algebras were constructed by Jankov in 1963. These are now called Jankov formulas. The Jankov formula of A is refuted on a Heyting algebra B iff A is a subalgebra of a homomorphic image of B . The Jankov formula of A axiomatizes the greatest variety that does not contain A . 2 / 41
De Jongh formulas De Jongh (1968) introduced what we now call de Jongh formulas (also called frame formulas). They are defined for finite rooted intuitionistic Kripke frames. The de Jongh formula of a frame F is refuted on a frame G iff F is a p-morphic image of a generated subframe of G We will see that via Esakia duality Jankov formulas are dual to de Jongh formulas ⇒ Esakia Duality de Jongh formulas Jankov formulas ⇐ 3 / 41
Fine and Rautenberg formulas Fine (1974) defined what we now call Fine formulas (also frame formulas or Jankov-Fine formulas) for rooted transitive frames. They have the same property as de Jongh formulas expressed in terms of p-morphisms and generated subframes. Rautenberg (1980) gave a Jankov style purely algebraic description of these formulas in terms of modal algebra subalgebras and homomorphisms. ⇒ Modal duality Fine formulas Jankov-Rautenberg formulas ⇐ 4 / 41
Canonical formulas The formulas discussed so far axiomatize large classes of logics. However, there exist intermediate and transitive modal logics that are not axiomatized by these formulas. Zakharyaschev (1992) introduced canonical formulas and showed that every intermediate and transitive modal logic is axiomatized by canonical formulas. Canonical formulas are defined for a finite rooted frame F and a set D of antichains F . Unlike de Jongh formulas and Fine formulas in canonical formulas Zakharyaschev uses partial p-morphisms and the so called closed domain condition (CDC). 5 / 41
Canonical formulas The aim of this talk is to fill in question marks in the following diagram (for both intermediate and modal logics). ⇒ ?? duality canonical formulas ?? formulas ⇐ This will give a purely algebraic account of canonical formulas. In the intuitionistic case the algebraic side of the duality will be the category of Heyting algebras and ( ∧ , → ) -preserving maps. In the modal case the algebraic side will be the category of modal algebras and relativized homomorphisms. 6 / 41
Jankov-Rautenberg formulas Let A be a finite subdirectly irreducible (s.i.) Heyting algebra. Recall that A is s.i. iff there exists the second largest element s of A ; that is, there exists s ∈ A such that s is the largest element of A − { 1 } . For each a ∈ A Jankov introduced a new variable p a and defined the Jankov formula χ ( A ) as follows: [ � { p a ∧ b ↔ p a ∧ p b : a , b ∈ A }∧ χ ( A ) = � { p a ∨ b ↔ p a ∨ p b : a , b ∈ A }∧ � { p a → b ↔ p a → p b : a , b ∈ A }∧ � { p ¬ a ↔ ¬ p a : a ∈ A } ] → p s 7 / 41
Jankov-Rautenberg formulas Let A be a finite subdirectly irreducible K4 -algebra, H = � + ( A ) , and t be the second largest element of H . We recall that the Jankov-Rautenberg formula of A is χ ( A ) = � + [ � { p a ∨ b ↔ p a ∨ p b : a , b ∈ A }∧ � { p a ∧ b ↔ p a ∧ p b : a , b ∈ A }∧ � { p ¬ a ↔ ¬ p a : a ∈ A }∧ � { p ♦ a ↔ ♦ p a : a ∈ A } ] → p t Jankov’s Theorem . For each (Heyting or modal) algebra B we have B �| = χ ( A ) iff A ∈ HS ( B ) . 8 / 41
Fine-De Jongh formulas Fine-de Jongh Theorem . For each finite rooted Kripke frame F there exists a formula J ( F ) such that for each frame G we have G �| = J ( F ) iff F is a p -morphic image of a generated subframe of G . We will link Jankov-Rautenberg formulas and Fine-de Jongh formulas via Esakia duality. 9 / 41
Esakia Duality Duality theory for Heyting and S4 -algebras was developed by Esakia (1934-2010) in 1970s. This provides the bridge between the algebraic and model-theoretic technique. The bridge is topological in nature. Recall that a modal space is a triple ( X , τ, R ) , where: ( X , τ ) is a compact, Hausdorff, zero-dimensional space. 1 R is a binary relation. 2 R ( x ) is closed for each x ∈ X . Here R ( x ) = { y ∈ X : xRy } . 3 If U is clopen (closed and open), then so is � R � U . Here 4 � R � U = { x ∈ X : R ( x ) ∩ U � = ∅} . If R is a partial order, then we call ( X , τ, R ) an Esakia space. 10 / 41
Esakia Duality Also recall that given two modal (Esakia) spaces X and Y , a map f : X → Y is a modal (Esakia) morphism if: f is continuous. 1 f is a p -morphism; that is, 2 xRz implies f ( x ) Rf ( z ) for each x , z ∈ X . For each x ∈ X and y ∈ Y , if f ( x ) Ry , then there exists z ∈ X such that xRz and f ( z ) = y . 11 / 41
Let MS ( Esa ) denote the category of modal (Esakia) spaces and modal (Esakia) morphisms. Let also MA ( Heyt ) denote the category of modal (Heyting) algebras and modal (Heyting) algebra homomorphisms Theorem: (Esakia, 1974) Heyt is dually equivalent to Esa . MA is dually equivalent to MS . For (a modal or Heyting) algebra A we let A ∗ be its dual space. For (a modal or Esakia) space X we let X ∗ be its dual algebra. 12 / 41
Duality ⇒ Duality Generated subframes Homomorphic images ⇐ ⇒ Duality p-morphic images Subalgebras ⇐ ⇒ Duality de Jongh formulas Jankov formulas ⇐ ⇒ Duality Fine form. Jankov-Rautenberg form. ⇐ 13 / 41
Generalized Esakia Duality Now let A and B be two Heyting algebras and h : A → B be a map. In order for h to be a morphism of Heyt , h should preserve all Heyting algebra operations. It is useful to work with such h : A → B which only preserve ∧ , → , and 1. In fact, as long as h preserves → , it automatically preserves 1! Therefore, we are interested in ( ∧ , → ) -homomorphisms; that is, those h : A → B for which we have h ( a ∧ b ) = h ( a ) ∧ h ( b ) and h ( a → b ) = h ( a ) → h ( b ) . 14 / 41
Generalized Esakia Duality Let h : A → B be a ( ∧ , → ) -homomorphism. If h ( 0 ) = 0, then we call h a ( ∧ , → , 0 ) -homomorphism; and if h ( a ∨ b ) = h ( a ) ∨ h ( b ) , then we call h a ( ∧ , → , ∨ ) -homomorphism. Clearly h is a Heyting algebra homomorphism iff h is a ( ∧ , → , ∨ , 0 ) -homomorphism. This outlook provides us with three new categories: Heyt ( ∧ , → ) is the category of Heyting algebras and 1 ( ∧ , → ) -homomorphisms. Heyt ( ∧ , → , 0 ) is the category of Heyting algebras and 2 ( ∧ , → , 0 ) -homomorphisms. Heyt ( ∧ , → , ∨ ) is the category of Heyting algebras and 3 ( ∧ , → , ∨ ) -homomorphisms. 15 / 41
Generalized Esakia Duality Given Heyting algebras A and B , we need to give the dual description of a ( ∧ , → ) -homomorphism h : A → B . It turns out that ( ∧ , → ) -homomorphisms h : A → B can dually be characterized by special partial maps from the Esakia dual B ∗ of B to the Esakia dual A ∗ of A . Let h : A → B be a ( ∧ , → ) -homomorphism. Unlike the case of Heyting algebra homomorphisms, the inverse image h − 1 ( x ) of a prime filter x of B may not be a prime filter of A . And this is exactly why the dual of h is a partial function! 16 / 41
Generalized Esakia Duality Let X and Y be Esakia spaces and f : X → Y a partial map. Let dom ( f ) denote the domain of f . We call f a partial Esakia morphism provided that: If x , z ∈ dom ( f ) and x ≤ z , then f ( x ) ≤ f ( z ) . 1 If x ∈ dom ( f ) , y ∈ Y , and f ( x ) ≤ y , then there exists 2 z ∈ dom ( f ) such that x ≤ z and f ( z ) = f ( y ) . x ∈ dom ( f ) iff f [ ↑ x ] = ↑ y for some y ∈ Y . 3 f [ ↑ x ] is a closed subset of Y . 4 If U is a clopen upset of Y , then X − ↓ f − 1 ( Y − U )) is a 5 clopen upset of X . 17 / 41
Generalized Esakia Duality Let Esa P denote the category of Esakia spaces and partial Esakia morphisms. Note that the composition of partial Esakia morphisms is not a composition of the partial maps. Theorem: Heyt ( ∧ , → ) is dually equivalent to Esa P . 18 / 41
Subframe formulas Let A be a finite subdirectly irreducible (s.i.) Heyting algebra with the second largest element s . α ( A ) = [ � { p a ∧ b ↔ p a ∧ p b : a , b ∈ A }∧ � { p a → b ↔ p a → p b : a , b ∈ A } ] → p s ⇒ Gen ES Duality subframe formulas ( ∧ , → ) -Jankov formulas ⇐ Theorem . A HA B �| = α ( A ) iff there exists a homomorphic image C of B 1 and a 1-1 ( ∧ , → ) -embedding of A into C . An ES Y �| = α ( A ) iff there exists a generated subframe Z of Y 2 and a partial Esakia morphism from Y onto A ∗ . 19 / 41
Cofinal morphisms A subset Z of a poset X is called cofinal if for each x ∈ X there is y ∈ Y such that xRy . A partial map f : X → Y is called cofinal if dom ( f ) is cofinal. 20 / 41
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