Recent Results in Cohesive Powers R. Dimitrov Department of - - PowerPoint PPT Presentation

Recent Results in Cohesive Powers

• R. Dimitrov

Department of Mathematics and Philosophy, Western Illinois University

WDCM-2020 July 24, 2020

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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

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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

• r 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 Bn. (Soare) If M1 and M2 are maximal sets then there is an automorphism g of E∗ such that g(M1) = M2 (Soare) If Q1 and Q2 are rank-n quasimaximal sets, then there is an automorphism g of E∗ such that g(Q1) = Q2

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The Lattice L(V∞).

V∞ is a ℵ0-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∞). V1 ∨ V2 = cl(V1 ∪ V2) and V1 ∧ V2 = V1 ∩ V2.

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The Lattice L∗(V∞)

V1 =∗ V2 if V1 and V2 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∞.

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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, ↑) ∼ = Bn Yet L∗(V , ↑) is not always isomorphic to Bn.

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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 Bn, or (2) the lattice L(n, Qa) of all subspaces of an n-dimensional vector space over a certain extension Qa ( Q) of the field Q, or (3) a finite product of lattices from (1) and (2).

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Example 1: typeΩ(B) = ((1, 1, 1), (a, b, c)),

Then L∗(cl(B), ↑) ∼ = the Boolean algebra B3 V∞ cl(B)

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Example 2: typeΩ(B1) = (3, (a)), and typeΩ(B2) = (3, (b))

Assume a = b. Then Qa ≇ Qb (By the FTPG) L∗(cl(B1), ↑) ≇ L∗(cl(B2), ↑)

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Cohesive powers of computable structures

Example: The cohesive power

C

Q of Q over C (1) Domain of

C

Q : {[ϕ]C | ϕ : ω → Q is a partial computable function, and C ⊆∗ dom(ϕ)} Here ϕ1 =C ϕ2 iff C ⊆∗ {x : ϕ1(x) ↓= ϕ2(x) ↓}. [ϕ]C is the equivalence class of ϕ with respect to =C (2) Pointwise operations: [ϕ1]C + [ϕ2]C = [ϕ1 + ϕ2]C and [ϕ1]C · [ϕ2]C = [ϕ1 · ϕ2]C (3) [0]C and [1]C are the equivalence classes of the corresponding total constant functions.

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[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

C

A, is a structure B for L with domain B such that the following holds. The set B = (D/ =C), where D = {ϕ | ϕ : ω → A is a partial computable function, and C ⊆∗ dom(ϕ)}. For ϕ1, ϕ2 ∈ D, we have ϕ1 =C ϕ2 iff C ⊆∗ {x : ϕ1(x) ↓= ϕ2(x) ↓}. If f ∈ L is an n-ary function symbol, then f Bis an n-ary function on B such that for every [ϕ1], . . . , [ϕn] ∈ B, we have f B([ϕ1], . . . , [ϕn]) = [ϕ], where for every x ∈ A, ϕ(x) ≃ f A(ϕ1(x), . . . , ϕn(x)).

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Definition: Cohesive Powers of Computable Structures

If P ∈ L is an m-ary predicate symbol, then PB is an m-ary relation

• n B such that for every [ϕ1], . . . , [ϕm] ∈ B,

PB([ϕ1], . . . , [ϕm]) iff C ⊆∗ {x ∈ A | PA(ϕ1(x), . . . , ϕm(x))}. If c ∈ L is a constant symbol, then cB is the equivalence class of the (total) computable function on A with constant value cA.

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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

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Theorem

(Fundamental theorem of cohesive powers)

1 If τ(y1, . . . , yn) is a term in L and [ϕ1], . . . , [ϕn] ∈ B, then

[τ B([ϕ1], . . . , [ϕn])] is the equivalence class of a p.c. function such that τ B([ϕ1], . . . , [ϕn])(x) = τ A(ϕ1(x), . . . , ϕn(x)).

2 If Φ(y1, . . . , yn) is a formula in L that is a boolean combination of Σ1

and Π1 formulas and [ϕ1], . . . , [ϕn] ∈ B, then B | = Φ([ϕ1], . . . , [ϕn]) iff R ⊆∗ {x : A | = Φ(ϕ1(x), . . . , ϕn(x))}.

3 If Φ is a Π3 sentence in L, then B |

= Φ implies A | = Φ.

4 If Φ is a Π2 (or Σ2) sentence in L, then B |

= Φ iff A | = Φ.

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Observations

If A is: (Dense) Linear Order (Without Endpoints), Ring, (Algebraically Closed) Field , Lattice, (Atomless) Boolean Algebra, then so is

C

A.

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• bservations

Let B =

C

A be the cohesive power of A. For c ∈ A let [ϕc] ∈ B be the equivalence class of ϕc such that ϕc(i) = c. The map d : A → B such that d(c) = [ϕc] is called the canonical embedding of A into B.

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Properties:

If A is finite then

C

A ∼ = A. if there is a computable permutation σ of ω such that σ(C1) =∗ C2, then

C1

A ∼ =

C2

A. If A1 ∼ = A2 are computably isomorphic then

C

A1∼ =

C

A2. Let C be be co-r.e. Then for every [ϕ] ∈

C

A there is a computable function f such that [f ] = [ϕ]. f (n) = ϕ(n) if ϕ(n) ↓ first, a if n is enumerated into C first.

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Limited Los for

M

N ∼ = R/ ∼M

If M is a maximal set, then

M

N ∼ = R/ ∼M Feferman-Scott-Tennenbaum [1959]: R/ ∼A is a model of only a fragment of First-Order-Arithmetics. N | = ∀x∃s∀e ≤ x[ϕe(x) ↓→ ϕe,s(x) ↓] but R/ ∼A ∀x∃s∀e ≤ x[ϕe(x) ↓→ ϕe,s(x) ↓]

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Results by J.Hirschfeld, Lerman, McLaughlin, Wheeler

Theorem: (Lerman 1975) (1) If A1 ≡m A2 are r-maximal sets, then R/A1 ∼ = R/A2. (2) If M1 and M2 are maximal sets of different m-degrees, then R/M1 and R/M2 are not even elementary equivalent. Theorem: (Hirschfeld and Wheeler 1975), (McLaughlin 1990) (3) R/M is rigid

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Dimitrov, Harizanov, Miller, Mourad 2014

1 If M1 ≡m M2, then

M1

Q ∼ =

M2

Q.

2 If M1 ≡m M2, then

M1

Q ≡

M2

Q.

3

M

Z is rigid.

4

M

Q is rigid. Qa is the cohesive power of Q with respect to the complement of a maximal set with m-degree a (4) Qa is rigid. (2) If a = b, then Qa is not isomorphic to Qb.

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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.

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Definability of N in Q

N is definable in Z : y ∈ N ⇔ ∃z1 . . . ∃z4[y = z2

1 + z2 2 + z2 3 + z2 4]

Corollary: N definable in Q: x ∈ N ⇔ ∃y1 . . . ∃y4[

i≤4 yi ∈ Z ∧ x = y2 1 + y2 2 + y2 3 + y2 4 ]

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Dimitrov, Harizanov, Orbits of Maximal Vector Spaces, Algebra and Logic, 2015.

There is an automorphism σ of L∗(V∞) such that σ(cl(B1)) = cl(B2) iff type(B1) = type(B2).

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Example 1: typeΩ(B) = ((1, 1, 1), (a, b, c)),

Then L∗(cl(B), ↑) ∼ = the Boolean algebra B3 V∞ cl(B)

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Example 2: typeΩ(B1) = (3, (a)), and typeΩ(B2) = (3, (b))

Assume a = b. Then Qa ≇ Qb (By the FTPG) L∗(cl(B1), ↑) ≇ L∗(cl(B2), ↑)

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Main Results: Cohesive Powers of Linear Orders, Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, Vatev, 2019

If A is a computable presentation of the linear order N with a computable successor function, then

• C

A ∼ =N + Q × Z. There is a computable presentation B of the linear order N with a non-computable successor function, s.t.

• C

B ∼ =N + Q × Z

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Main Results: Cohesive Powers of Linear Orders, Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, Vatev, 2019

There is a computable presentation D of the linear order N, s.t.

• C

D ≇N + Q × Z In fact, there is an element of

C

D that has no immediate successor. Let ϕ be a 0

3 sentence that states that every element has an

immediate sucessor. Then N ϕ while

• C

D ϕ

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References

[1] Davis M., Matiyasevich Y., and Robinson J., Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974),

• Amer. Math. Soc., Providence, RI, pp. 323–378 (1976).

[2] Dimitrov, R.D.: Quasimaximality and principal filters isomorphism between E∗ and L∗(V∞), Arch. Math. Logic 43, pp. 415–424 (2004). [3] Dimitrov, R.D.: A class of Σ0

3 modular lattices embeddable as principal

filters in L∗(V∞), Arch. Math. Logic 47, no. 2, pp. 111–132 (2008). [4] Dimitrov, R.D.: Cohesive powers of computable structures, Annuare De L’Universite De Sofia “St. Kliment Ohridski”, Fac. Math and Inf., tome 99, pp. 193–201 (2009). [5a] Dimitrov, R., Harizanov, V., Miller, R., and Mourad, K.J.: Isomorphisms on non-standard fields and Ash’s conjecture. Language, Life, Limits, Lecture Notes in Comput. Sci., 8493, pp. 143–152, Springer, Cham, (2014).

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References

[5b] Dimitrov, R.D. and Harizanov, V.: Orbits of maximal vector spaces, Algebra and Logic 54 (2015), pp. 680–732 (Russian); (2016) pp. 440–477 (English translation). [5c] Dimitrov, Harizanov,Morozov, Shafer, A. Soskova, Vatev, Cohesive Powers of Linear Orders, in CiE2019 proceedings. [6] Feferman, S., Scott, D.S., and Tennenbaum, S.: Models of arithmetic through function rings, Notices Amer. Math. Soc. 6, 173. Abstract #556-31 (1959). [7] Guichard, D.R.: Automorphisms of substructure lattices in recursive algebra, Ann. Pure Appl. Logic, vol. 25, no. 1, pp. 47–58 (1983). [8] Hirschfeld, J.: Models of arithmetic and recursive functions, Israel Journal of Mathematics, vol. 20, no.2, pp. 111–126 (1975).

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References

[9] Hirschfeld, J. and Wheeler, W.: Forcing, arithmetic, division rings, Lecture Notes in Mathematics, vol. 454, Springer, Berlin (1975). [10] Robinson, J.: Definability and decision problems in arithmetic, Journal

• f Symbolic Logic, vol. 14, no. 2, pp. 98–114 (1949).

[11] Koenigsmann, J.: Defining Z in Q, forthcoming in the Annals of

• Mathematics. (http://arxiv.org/abs/1011.3424)

[12] Lerman, M.: Recursive functions modulo co-r-maximal sets, Transactions of the American Mathematical Society, vol. 148, no. 2 , pp. 429–444 (1970).

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References

[13] McLaughlin, T.: Some extension and rearrangement theorems for Nerode semirings, Zeitschr. f. math. Logic und Grundlagen d. Math., vol. 35, pp. 197–209 (1989). [14] McLaughlin, T.: Sub-arithmetical ultrapowers: a survey, Annals of Pure and Applied Logic, vol. 49, no 2, pp. 143–191 (1990). [15] McLaughlin, T.: ∆1 Ultrapowers are totally rigid, Archive for Mathematical Logic, vol. 46, pp. 379–384 (2007). [16] Metakides, G. and Nerode, A.: Recursively enumerable vector spaces, Annals of Mathematical Logic, vol. 11, pp. 147–171 (1977). [17] Robinson, R.: Arithmetical definitions in the ring of integers, Proceedings of the American Mathematical Society, vol. 2, no. 2, pp. 279–284 (1951). [18] Soare, R.I.: Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin (1987).

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Thank you

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