Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Contents Epimorphism surjectivity and the Beth definability property 1. Beth and epimorphisms Tommaso Moraschini Joint with: Guram Bezhanishvili and James Raftery 2. Blok-Hoogland’s conjecture 3. Finite depth September 25, 2017 1 / 18 2 / 18 Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Epimorphism surjectivity Beth property ◮ Let L be algebraizable with equivalence formulas ρ ( x , y ) . Definition Definition Let K be a class of algebras. A homomorphism f : A → B in K is an Let Γ be a set of formulas over X ∪ Z with X ∩ Z = ∅ and X � = ∅ . epimorphism if for every pair g , h : B ⇒ C of homomorphisms in K 1. Γ implicitly defines Z in terms of X if if g ◦ f = h ◦ f , then g = h . Γ ∪ σ ( Γ ) ⊢ L ρ ( z , σ z ) for every z ∈ Z ◮ Are epis surjective in a variety? for every substitution σ that fixes X . ◮ Yes: Boolean algebras, Heyting algebras, lattices, semilattices 2. Γ explicitly defines Z in terms of X if for every z ∈ Z there is and (Abelian) groups. a formula ϕ z over X only such that ◮ No: distributive lattices, rings with unity and monoids. Γ ⊢ L ρ ( z , ϕ z ) . ◮ Thus epimorphism surjectivity is not preserved in subvarieties! ◮ L has the (resp. finite) Beth property when 1 (resp. with Z finite) implies 2. 4 / 18 5 / 18
Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Transfer theorem Why Heyting algebras? ◮ We want to establish Blok and Hoogland’s conjecture by Definition finding a variety (that algebraizes a logic) where: A homomorphism f : A → B is almost onto if B is generated by 1. Almost onto epimorphisms are surjective. f ( A ) ∪ { b } for some b ∈ B . 2. Epimorphisms need not be surjective. Theorem (Kreisel) Theorem (Blok and Hoogland) Every axiomatic extension of IPC has the finite Beth property. Let L be an algebraizable logic. 1. L has the Beth property iff epis are surjective in Alg ∗ L . ◮ This result can be re-stated as follows: 2. L has the finite Beth property iff almost onto epis are Theorem surjective in Alg ∗ L . In varieties of Heyting algebras almost onto epis are surjective. ◮ Blok and Hoogland conjectured that ◮ To establish Blok and Hoogland’s conjecture, it is enough to Beth property � = finite Beth property. find a variety of Heyting algebras where epis need not be surjective. 6 / 18 7 / 18 Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth ◮ Let A be the Heyting algebra depicted below and B the K-epic subalgebras subalgebra with universe { 0 , b 0 , b 1 , b 2 , . . . , 1 } . Definition 1 Let K be a quasi-variety and B ∈ K. A subalgebra A ≤ B is K-epic r if for every pair of homomorphisms f , g : B ⇒ C ∈ K r � ❅ � ❅ b 0 c 0 r r ❅ r � ❅ � if f ↾ A = g ↾ A , then f = g . r � ❅ � ❅ b 1 c 1 r r ◮ Epis are surjective in K iff no B ∈ K has a proper K-epic ❅ r � ❅ � r � ❅ subalgebra. � ❅ b 2 c 2 r r ❅ ❅ r � � ❅ � Theorem (Campercholi) � ❅ � ❅ Let K be a quasi-variety and A ≤ B ∈ K. TFAE: 1. A is a K-epic subalgebra of B . 2. For every b ∈ B there is a primitive positive formula ϕ ( � x , y ) ◮ We claim that B is a V ( A ) -epic subalgebra of A . a ∈ A such that and � ◮ We need to find primitive positive formulas that define partial functions in V ( A ) and, moreover, construct A out of B . K � ∀ � x , y , z (( ϕ ( � x , y )& ϕ ( � x , z )) → y ≈ z ) and B � ϕ ( � a , b ) . 9 / 18 10 / 18
Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Partial functions Two Beth properties ◮ Consider the conjunction of equations ϕ ( x 0 , x 1 , x 2 , y 0 , y 1 , y 2 ) := & ◮ Epimorphisms need not to be surjective in V ( A ) . ( x n → y n ≈ y n & y n → x n ≈ x n ) ◮ Observe that V ( A ) satisfies the weak Pierce law n ≤ 2 & ( x n ∧ y n ≈ x n + 1 ∨ y n + 1 ) . ( y → x ) ∨ ((( x → y ) → x ) → x ) ≈ 1 . n ≤ 1 ◮ and the primitive positive formula ◮ Then V ( A ) is locally finite. Φ( x 0 , x 1 , x 2 , y 0 ) := ∃ y 1 y 2 ϕ. Theorem (Blok-Hoogland’s conjecture) ◮ Φ define a partial 3-ary function in V ( A ) : For every C ∈ V ( A ) 1. Epimorphisms need not be surjective in locally finite varieties and a 0 , a 1 , a 2 ∈ C there is at most one e ∈ C s.t. of Heyting algebras. C � Φ( a 0 , a 1 , a 2 , e ) . 2. The Beth property and the finite Beth property are different in locally tabular superintuitionistic logics. ◮ Applying this partial function to B we recover the whole A : A � Φ( b n + 2 , b n + 1 , b n , c n ) for every n ∈ ω. 11 / 18 12 / 18 Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Rieger-Nishimura lattice Finite depth Definition Definition A Heyting algebra A has depth n if the longest chain in � Pr ( A ) , ⊆� A Heyting algebra A has width n if the largest antichain in principal has exactly n elements. upsets of � Pr ( A ) , ⊆� has exactly n elements. Let HA n be the class of Heyting algebras of depth ≤ n . ◮ Let W n be the class of Heyting algebras of width ≤ n . It is a Theorem (Maksimova and Ono) variety. ◮ V ( A ) has width 2. HA n is a variety axiomatized by h n ≈ 1, where h 0 = y and for n > 0 ◮ The Rieger-Nishimura lattice has width 2. h n := x n ∨ ( x n → h n − 1 ) . Theorem ◮ A variety of Heyting algebras has finite depth when its 1. There is a continuum of varieties of Heyting algebras width members have finite depth. ≤ 2 where epimorphisms need not be surjective. Theorem 2. Among them there is the variety generated by the Let K be a variety of Heyting algberas. If K has finite depth, then Rieger-Nishimura lattice. epimorphisms are surjective in K. 13 / 18 15 / 18
Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Consequences Strong epimorphism surjectivity Definition A class of algebras K has strong epimorphism surjectivity if ◮ Finitely generated varieties of Heyting algebras are known to whenever f : A → B is homomorphism in K and b ∈ B � f ( A ) , have finite depth. there are homomorphisms g , h : B ⇒ C in K such that Corollary g ◦ f = h ◦ f and g ( b ) � = h ( b ) . 1. Epimorphisms are surjective in finitely generated varieties of Heyting algberas. Theorem (Maksimova) 2. Tabular superintuitionistic logics have the Beth property. There are finitely many varieties of Heyting algebras with strong 3. Superintuitionistic logics, whose theorems include h n for some epimorphism surjectitiy. n ∈ ω , have the Beth property. 4. Epimorphisms are surjective in all varieties of Gödel algebras. ◮ There is a continuum of varieties of depth ≤ 3. ◮ Thus there is a continuum of varieties with epimorphism surjectivity but not strong epimorphism surjectivity. 16 / 18 17 / 18 Beth and epimorphisms Blok-Hoogland’s conjecture Finite depth Thanks for coming! 18 / 18
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