Enrichable Elements in Heyting Algebras Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras Logic KM Kuznetsov’s Theorem KM-algebras Alexei Muravitsky Enrichable alexeim@nsula.edu Heyting algebras Enriched elements Kuznetsov’s Theorem revisited Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Outline Enrichable Elements in Heyting Algebras 1 Logic KM Alexei Kuznetsov’s Theorem Muravitsky alexeim@nsula.edu KM-algebras Logic KM Kuznetsov’s 2 Enrichable Heyting algebras Theorem KM-algebras Enriched elements Enrichable Kuznetsov’s Theorem revisited Heyting algebras Enriched elements Kuznetsov’s Theorem revisited Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Outline Enrichable Elements in Heyting Algebras 1 Logic KM Alexei Kuznetsov’s Theorem Muravitsky alexeim@nsula.edu KM-algebras Logic KM Kuznetsov’s 2 Enrichable Heyting algebras Theorem KM-algebras Enriched elements Enrichable Kuznetsov’s Theorem revisited Heyting algebras Enriched elements Kuznetsov’s 3 Embedding Theorem revisited E -completion Embedding Simple completions E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Outline Enrichable Elements in Heyting Algebras 1 Logic KM Alexei Kuznetsov’s Theorem Muravitsky alexeim@nsula.edu KM-algebras Logic KM Kuznetsov’s 2 Enrichable Heyting algebras Theorem KM-algebras Enriched elements Enrichable Kuznetsov’s Theorem revisited Heyting algebras Enriched elements Kuznetsov’s 3 Embedding Theorem revisited E -completion Embedding Simple completions E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Logic KM Enrichable Elements in Heyting Algebras Alexei Muravitsky Propositional languages L and L − : alexeim@nsula.edu an infinite set of propositional variables p , q , . . . ; Logic KM Kuznetsov’s connectives: ∧ , ∨ , → , ¬ (assertoric connectives) and � (a Theorem KM-algebras unary modality); Enrichable Heyting L is the full language above, L − is the assertoric part of L . algebras Formulas in L − are denoted by letters A , B , . . . . Enriched elements Kuznetsov’s Theorem revisited Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Logic KM Enrichable Elements in Heyting Algebras Alexei Muravitsky KM is Int understood in language L plus the following alexeim@nsula.edu formulas as axioms: Logic KM Kuznetsov’s • p → � p Theorem KM-algebras • ( � p → p ) → p Enrichable Heyting • � p → ( q ∨ ( q → p )) algebras Enriched elements closed under substitution and detachment ( modus ponens ). Kuznetsov’s Theorem revisited Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Kuznetsov’s Theorem Enrichable Elements in Heyting Theorem (Kuznetsov’s Theorem) Algebras Alexei For any formulas A and B of L − , Muravitsky alexeim@nsula.edu Int + A ⊢ B ⇔ KM + A ⊢ B . Logic KM Kuznetsov’s Theorem KM-algebras Why is KM interesting? Enrichable Heyting algebras Why is Kuznetsov’s Theorem interesting? Enriched elements Kuznetsov’s KM nowadays is mentioned in connection with Lax Theorem revisited (Fairtlough-Mendler) or mHC (Esakia). However, having been Embedding defined in the end of the 1970s, it stemmed from a different E -completion Simple completions source. Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Kuznetsov’s Theorem Enrichable The following diagram is commutative: Elements in Heyting Algebras Alexei Muravitsky κ alexeim@nsula.edu Ext GL > Ext KM Logic KM Kuznetsov’s Theorem µ λ KM-algebras Enrichable ∨ ∨ σ Heyting Ext Grz > Ext Int algebras Enriched elements Kuznetsov’s Here κ is a lattice isomorphism (Muravitsky), λ is a meet Theorem revisited epimorphism (Kuznetsov’s Theorem) and µ is also a meet Embedding epimorphism (Keznetsov-Muravitsky). E -completion Simple This in particular implies that any intermediate logic is the completions superintuitionistic fragment of some GL -logic. Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
KM-algebras Enrichable Elements in Heyting A = ( A , ∧ , ∨ , ¬ , 0 , 1 , � ), where ( A , ∧ , ∨ , ¬ , 0 , 1 ) is a Heyting Algebras algebra (the Heyting reduct of A ) and � is subject to the Alexei Muravitsky following conditions (identities): alexeim@nsula.edu Logic KM • � x ≤ x Kuznetsov’s • � x → x = x Theorem KM-algebras • � x ≤ y ∨ ( y → x ) Enrichable Heyting algebras Enriched Theorem (algebraic version of Kuznetsov’s Theorem) elements Kuznetsov’s Theorem Any Heyting algebra can be embedded into a KM -algebra such revisited Embedding that the Heyting reduct of the latter generates the same variety E -completion as the initial algebra. Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Enrichable Heyting Algebras Enrichable Elements in Heyting Algebras Alexei Muravitsky alexeim@nsula.edu In the remaining part of the this presentation, Logic KM Kuznetsov’s A will denote a Heyting algebra. Theorem KM-algebras Enrichable Heyting algebras Question: In how many ways can one make a Heyting algebra Enriched elements a KM -algebra? Kuznetsov’s Theorem revisited Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Enriched Elements Enrichable Elements in Heyting Algebras Definition Alexei Muravitsky Given algebra A and its elements a and a ∗ , the pair ( a , a ∗ ) is alexeim@nsula.edu called an E -pair if the following (in)equalities hold: Logic KM Kuznetsov’s Theorem • a ≤ a ∗ KM-algebras Enrichable • a ∗ → a = a Heyting a ∗ ≤ b ∨ ( b → a ) , for any b ∈ A . algebras • Enriched elements Kuznetsov’s If ( a , a ∗ ) is an E -pair, we say that a is enriched by a ∗ in A , or Theorem revisited a ∗ enriches a. Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
Enrichable Heyting Algebras Enrichable Elements in Observation Heyting If ( a , a ′ ) and ( a , a ′′ ) are E -pairs of A then a ′ = a ′′ . Algebras Alexei Muravitsky Corollary alexeim@nsula.edu There may be only one way to make a Heyting algebra a Logic KM KM -algebra, if each element of the former is enrichable. Kuznetsov’s Theorem KM-algebras Definition Enrichable Heyting An algebra is called enrichable if each element of it is algebras Enriched enrichable. elements Kuznetsov’s Theorem revisited Theorem (Kuznetsov’s Theorem revisited) Embedding E -completion Every Heyting algebra is embedded into an enrichable algebra Simple completions such that the latter and the former generate the same variety. Question: How can such an embedding be done? Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
E -completion Enrichable Elements in Heyting Algebras Let us fix this notation: Alexei Muravitsky an (initial) algebra A , alexeim@nsula.edu ( µ A , ⊆ ), the poset of the prime filters of A , Logic KM H ( A ), the Heyting algebra of the upward cones over Kuznetsov’s Theorem KM-algebras ( µ A , ⊆ ), Enrichable Heyting h : A →H ( A ), Stone embedding, algebras △ X = { F ∈ µ A | ∀ F ′ ( F ⊂ F ′ ⇒ F ′ ∈ X ) } , Enriched elements Kuznetsov’s Theorem B △ ( A ) is the subalgebra of H ( A ) generated by revisited { h ( a ) | a ∈ A } ∪ { △ h ( a ) | a ∈ A } . Embedding E -completion Simple completions Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
E -completion Enrichable Elements in Definition ( E -completion) Heyting Algebras We first define the following sequence of algebras: Alexei A 0 = A , Muravitsky alexeim@nsula.edu A i +1 = B △ ( A i ) , i < ω . Logic KM Next we observe that { A i } i <ω is a direct family of algebras and Kuznetsov’s Theorem → KM-algebras define A to be the direct limit of { A i } i <ω . Enrichable Heyting algebras Observation Enriched elements We observe the following: Kuznetsov’s Theorem → revisited A belongs to the variety generated by all A i . Embedding E -completion → Simple If A is subdirectly irreducible, then all A i and A are completions subdirectly irreducible as well. Alexei Muravitsky alexeim@nsula.edu Enrichable Elements in Heyting Algebras
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