Recent Results in Cohesive Powers R. Dimitrov Department of Mathematics and Philosophy, Western Illinois University WDCM-2020 July 24, 2020 RD WDCM-2020 07/27/2020 1 / 32
Content Definitions Motivation: [Dimitrov, 2004, 2008, 2009] Cohesive powers used in structural theorems about L ∗ ( V ∞ ). Recent Results: [Dimitrov, Harizanov, Miller, Mourad, 2014] Isomorphisms on non-standard fields and Ash’s conjecture [Dimitrov, Harizanov, 2015] Orbits of maximal vector spaces [Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, Vatev, 2019] Cohesive Powers of Linear Orders RD WDCM-2020 07/27/2020 2 / 32
Definitions from Computability Theory E is the lattice of computably enumerable sets. E ∗ is the lattice E modulo finite sets. An infinite set C ⊂ ω is cohesive if for every c.e. set W either W ∩ C or W ∩ C is finite. A set M is maximal if M is c.e. and M is cohesive. Theorem: [Friedberg 1958] There are maximal sets. B is quasimaximal if it is the intersection of finitely many ( n ≥ 1) maximal sets. E ∗ ( B , ↑ ) is isomorphic to the Boolean algebra B n . (Soare) If M 1 and M 2 are maximal sets then there is an automorphism g of E ∗ such that g ( M 1 ) = M 2 (Soare) If Q 1 and Q 2 are rank- n quasimaximal sets, then there is an automorphism g of E ∗ such that g ( Q 1 ) = Q 2 RD WDCM-2020 07/27/2020 3 / 32
The Lattice L ( V ∞ ). V ∞ is a ℵ 0 -dimensional vector space over a computable field F . The vectors in V ∞ are finite sequences of elements of Q . If I is a set of vectors from V ∞ , then cl ( I ) denotes the linear span of I . V ⊆ V ∞ is computably enumerable if the set of vectors in V is c.e. The c.e. subspaces of V ∞ form a lattice, denoted by L ( V ∞ ). V 1 ∨ V 2 = cl ( V 1 ∪ V 2 ) and V 1 ∧ V 2 = V 1 ∩ V 2 . RD WDCM-2020 07/27/2020 4 / 32
The Lattice L ∗ ( V ∞ ) V 1 = ∗ V 2 if V 1 and V 2 differ by finite dimension. L ∗ ( V ∞ ) is L ( V ∞ ) / = ∗ . Both L ( V ∞ ) and L ∗ ( V ∞ ) are modular nondistributive lattices. There are very few result about the structure of L ∗ ( V ∞ ). Conjecture: (Ash) The automorphisms of L ∗ ( V ∞ ) are induced by computable semilinear transformations with finite dimensional kernels and co-finite dimensional images in V ∞ . RD WDCM-2020 07/27/2020 5 / 32
The map Cl : E ∗ → L ∗ ( V ∞ ) Let A be a computable basis of V ∞ and let B be a quasimaximal subset of A . Let V = cl ( B ) . Identify E ∗ with the lattice of c.e. subsets of A modulo = ∗ . E ∗ ( B , ↑ ) ∼ = B n Yet L ∗ ( V , ↑ ) is not always isomorphic to B n . RD WDCM-2020 07/27/2020 6 / 32
Characterization of L ∗ ( V , ↑ ) Theorem(s): (Dimitrov 2004, 2008) Let B be a rank-n quasimaximal subset of a computable basis Ω of V ∞ Then L ∗ ( cl ( B ) , ↑ ) ∼ = (1) the Boolean algebra B n , or (2) the lattice L ( n , Q a ) of all subspaces of an n -dimensional vector space over a certain extension Q a ( � Q ) of the field Q , or (3) a finite product of lattices from (1) and (2). RD WDCM-2020 07/27/2020 7 / 32
Example 1: type Ω ( B ) = ((1 , 1 , 1) , ( a , b , c )) , Then L ∗ ( cl ( B ) , ↑ ) ∼ = the Boolean algebra B 3 V ∞ cl ( B ) RD WDCM-2020 07/27/2020 8 / 32
Example 2: type Ω ( B 1 ) = (3 , ( a )) , and type Ω ( B 2 ) = (3 , ( b )) Assume a � = b . Then Q a ≇ Q b (By the FTPG) L ∗ ( cl ( B 1 ) , ↑ ) ≇ L ∗ ( cl ( B 2 ) , ↑ ) RD WDCM-2020 07/27/2020 9 / 32
Cohesive powers of computable structures Example: The cohesive power � Q of Q over C C (1) Domain of � Q : C { [ ϕ ] C | ϕ : ω → Q is a partial computable function, and C ⊆ ∗ dom ( ϕ ) } Here ϕ 1 = C ϕ 2 iff C ⊆ ∗ { x : ϕ 1 ( x ) ↓ = ϕ 2 ( x ) ↓} . [ ϕ ] C is the equivalence class of ϕ with respect to = C (2) Pointwise operations: [ ϕ 1 ] C + [ ϕ 2 ] C = [ ϕ 1 + ϕ 2 ] C and [ ϕ 1 ] C · [ ϕ 2 ] C = [ ϕ 1 · ϕ 2 ] C (3) [0] C and [1] C are the equivalence classes of the corresponding total constant functions. RD WDCM-2020 07/27/2020 10 / 32
[D2008]: Cohesive Powers of Computable Structures Let A be a computable structure for a computable language L and with domain A , and let C ⊂ ω be a cohesive set. The cohesive power of A over C , denoted by � A , is a structure B for L with domain B such that the C following holds. The set B = ( D / = C ), where D = { ϕ | ϕ : ω → A is a partial computable function, and C ⊆ ∗ dom ( ϕ ) } . For ϕ 1 , ϕ 2 ∈ D , we have ϕ 1 = C ϕ 2 iff C ⊆ ∗ { x : ϕ 1 ( x ) ↓ = ϕ 2 ( x ) ↓} . If f ∈ L is an n -ary function symbol, then f B is an n -ary function on B such that for every [ ϕ 1 ] , . . . , [ ϕ n ] ∈ B , we have f B ([ ϕ 1 ] , . . . , [ ϕ n ]) = [ ϕ ], where for every x ∈ A , ϕ ( x ) ≃ f A ( ϕ 1 ( x ) , . . . , ϕ n ( x )) . RD WDCM-2020 07/27/2020 11 / 32
Definition: Cohesive Powers of Computable Structures If P ∈ L is an m -ary predicate symbol, then P B is an m -ary relation on B such that for every [ ϕ 1 ] , . . . , [ ϕ m ] ∈ B , C ⊆ ∗ { x ∈ A | P A ( ϕ 1 ( x ) , . . . , ϕ m ( x )) } . P B ([ ϕ 1 ] , . . . , [ ϕ m ]) iff If c ∈ L is a constant symbol, then c B is the equivalence class of the (total) computable function on A with constant value c A . RD WDCM-2020 07/27/2020 12 / 32
Related Notions Homomorphic images of the semiring of recursive functions have been studied as models of fragments of arithmetics by Fefferman, Scott, and Tennenbaum, Hirschfeld, Wheeler, Lerman, McLaughlin Let A be a specific r-cohesive set and let f and g be computable functions. Fefferman, Scott, and Tennenbaum defined a structure R / ∼ A For recursive f and g let f ∼ A g if A ⊆ ∗ { n : f ( n ) = g ( n ) } The domain of R / ∼ A consists of the equivalence classes of recursive functions modulo ∼ A RD WDCM-2020 07/27/2020 13 / 32
Theorem (Fundamental theorem of cohesive powers) 1 If τ ( y 1 , . . . , y n ) is a term in L and [ ϕ 1 ] , . . . , [ ϕ n ] ∈ B , then [ τ B ([ ϕ 1 ] , . . . , [ ϕ n ])] is the equivalence class of a p.c. function such that τ B ([ ϕ 1 ] , . . . , [ ϕ n ])( x ) = τ A ( ϕ 1 ( x ) , . . . , ϕ n ( x )) . 2 If Φ( y 1 , . . . , y n ) is a formula in L that is a boolean combination of Σ 1 and Π 1 formulas and [ ϕ 1 ] , . . . , [ ϕ n ] ∈ B , then = Φ([ ϕ 1 ] , . . . , [ ϕ n ]) iff R ⊆ ∗ { x : A | B | = Φ( ϕ 1 ( x ) , . . . , ϕ n ( x )) } . 3 If Φ is a Π 3 sentence in L, then B | = Φ implies A | = Φ . 4 If Φ is a Π 2 (or Σ 2 ) sentence in L, then B | = Φ iff A | = Φ . RD WDCM-2020 07/27/2020 14 / 32
Observations If A is: (Dense) Linear Order (Without Endpoints), Ring, (Algebraically Closed) Field , Lattice, (Atomless) Boolean Algebra, then so is � A . C RD WDCM-2020 07/27/2020 15 / 32
observations Let B = � A be the cohesive power of A . C For c ∈ A let [ ϕ c ] ∈ B be the equivalence class of ϕ c such that ϕ c ( i ) = c . The map d : A → B such that d ( c ) = [ ϕ c ] is called the canonical embedding of A into B . RD WDCM-2020 07/27/2020 16 / 32
Properties: A ∼ If A is finite then � = A . C if there is a computable permutation σ of ω such that σ ( C 1 ) = ∗ C 2 , A ∼ then � = � A . C 1 C 2 If A 1 ∼ A 1 ∼ = A 2 are computably isomorphic then � = � A 2 . C C Let C be be co-r.e. Then for every [ ϕ ] ∈ � A there is a computable C function f such that [ f ] = [ ϕ ]. � ϕ ( n ) if ϕ ( n ) ↓ first, f ( n ) = a if n is enumerated into C first. RD WDCM-2020 07/27/2020 17 / 32
N ∼ Limited Los for � = R / ∼ M M N ∼ If M is a maximal set, then � = R / ∼ M M Feferman-Scott-Tennenbaum [1959]: R / ∼ A is a model of only a fragment of First-Order-Arithmetics. N | = ∀ x ∃ s ∀ e ≤ x [ ϕ e ( x ) ↓→ ϕ e , s ( x ) ↓ ] but R / ∼ A � ∀ x ∃ s ∀ e ≤ x [ ϕ e ( x ) ↓→ ϕ e , s ( x ) ↓ ] RD WDCM-2020 07/27/2020 18 / 32
Results by J.Hirschfeld, Lerman, McLaughlin, Wheeler Theorem: (Lerman 1975) (1) If A 1 ≡ m A 2 are r-maximal sets, then R / A 1 ∼ = R / A 2 . (2) If M 1 and M 2 are maximal sets of different m-degrees, then R / M 1 and R / M 2 are not even elementary equivalent. Theorem: (Hirschfeld and Wheeler 1975), (McLaughlin 1990) (3) R / M is rigid RD WDCM-2020 07/27/2020 19 / 32
Dimitrov, Harizanov, Miller, Mourad 2014 Q ∼ 1 If M 1 ≡ m M 2 , then � = � Q . M 1 M 2 2 If M 1 �≡ m M 2 , then � Q �≡ � Q . M 1 M 2 3 � Z is rigid. M 4 � Q is rigid. M Q a is the cohesive power of Q with respect to the complement of a maximal set with m-degree a (4) Q a is rigid. (2) If a � = b , then Q a is not isomorphic to Q b . RD WDCM-2020 07/27/2020 20 / 32
Definability of Z in Q J. Robinson (1949): Z is ∀∃∀ definable in Q . Poonen (2009): Z is ∃∀ definable in Q . Koenigsmann (2015): Z is ∀ definable in Q . RD WDCM-2020 07/27/2020 21 / 32
Definability of N in Q N is definable in Z : y ∈ N ⇔ ∃ z 1 . . . ∃ z 4 [ y = z 2 1 + z 2 2 + z 2 3 + z 2 4 ] Corollary: N definable in Q : i ≤ 4 y i ∈ Z ∧ x = y 2 1 + y 2 2 + y 2 3 + y 2 x ∈ N ⇔ ∃ y 1 . . . ∃ y 4 [ � 4 ] RD WDCM-2020 07/27/2020 22 / 32
Dimitrov, Harizanov, Orbits of Maximal Vector Spaces, Algebra and Logic, 2015. There is an automorphism σ of L ∗ ( V ∞ ) such that σ ( cl ( B 1 )) = cl ( B 2 ) iff type ( B 1 ) = type ( B 2 ) . RD WDCM-2020 07/27/2020 23 / 32
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