Orders on Computable Torsion-Free Abelian Groups Asher M. Kach - PowerPoint PPT Presentation

Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12 th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24

Outline Classical Algebra Background 1 Computing a Basis 2 Computing an Order 3 With A Basis Without A Basis Open Questions 4 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24

Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = ( G : + , 0 ) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if ( ∀ x ∈ G )( ∀ n ∈ ω ) [ x � = 0 ∧ n � = 0 = ⇒ nx � = 0 ] . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24

Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup of Q ω . Definition The rank of a countable torsion-free abelian group G is the least cardinal κ such that G is a subgroup of Q κ . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24

Examples of Torsion-Free Abelian Groups Example Any subgroup G of Q is torsion-free and has rank one. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 5 / 24

Examples of Torsion-Free Abelian Groups Example Any subgroup G of Q is torsion-free and has rank one. Example The subgroup H of Q ⊕ Q (viewed as having generators b 1 and b 2 ) generated by b 1 , b 2 , and b 1 + b 2 2 So elements of H look like β 1 b 1 + β 2 b 2 + α b 1 + b 2 for β 1 , β 2 , α ∈ Z . 2 has rank two. Remark Note that b 1 2 and b 2 2 do not belong to H despite their sum b 1 + b 2 2 belonging to H . We will often abuse notation and write such things as 1 2 b 1 + 1 2 b 2 for b 1 + b 2 . 2 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 5 / 24

Outline Classical Algebra Background 1 Computing a Basis 2 Computing an Order 3 With A Basis Without A Basis Open Questions 4 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 6 / 24

The Motivating Theorem Definition Fix a group G = ( G : + , 0 ) . A set B ⊂ G (not containing 0) is a basis if it is a maximal linearly independent set (with coefficients in Z ). Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 7 / 24

The Motivating Theorem Definition Fix a group G = ( G : + , 0 ) . A set B ⊂ G (not containing 0) is a basis if it is a maximal linearly independent set (with coefficients in Z ). Theorem Every torsion-free abelian group has a basis. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 7 / 24

The Motivating Theorem Definition Fix a group G = ( G : + , 0 ) . A set B ⊂ G (not containing 0) is a basis if it is a maximal linearly independent set (with coefficients in Z ). Theorem Every torsion-free abelian group has a basis. Question Does this remain true in the effective setting? In other words, does every computable torsion-free abelian group admit a computable basis? Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 7 / 24

Basis Results (I) Proposition (Folklore (?)) Every computable torsion-free abelian group G has a basis B ⊂ G computable from 0 ′ . Proof. Enumerate G as { a i } i ∈ ω . Recursively determine if we should place a i ∈ B by checking whether a i is nonzero and linearly independent (over Z ) from { a 0 , . . . , a i − 1 } . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 8 / 24

Basis Results (I) Proposition (Folklore (?)) Every computable torsion-free abelian group G has a basis B ⊂ G computable from 0 ′ . Proof. Enumerate G as { a i } i ∈ ω . Recursively determine if we should place a i ∈ B by checking whether a i is nonzero and linearly independent (over Z ) from { a 0 , . . . , a i − 1 } . Theorem The following are equivalent (over RCA 0 ): ACA 0 . Every torsion-free abelian group has a basis. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 8 / 24

Basis Results (I) Proof. Note that the linear (in)dependence relation can be computed from a basis. Given elements a i 0 , . . . , a i n , write each as a linear combination of the basis elements. Determine linear (in)dependence using linear algebra. Thus, it suffices to construct a computable group G for which the linear (in)dependence relation computes 0 ′ . Let G be the computable presentation of Z ω with generators { g i } i ∈ ω . If i enters K at stage s , set g 2 i + 1 = s g 2 i . Then i ∈ K if and only if g 2 i and g 2 i + 1 are linearly dependent. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 9 / 24

Basis Results (II) Theorem (Dobritsa (1983)) Every computable torsion-free abelian group G has an isomorphic computable H admitting a computable basis. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 10 / 24

Basis Results (II) Theorem (Dobritsa (1983)) Every computable torsion-free abelian group G has an isomorphic computable H admitting a computable basis. Corollary Every computable torsion-free abelian group G of infinite rank has an isomorphic computable H for which every basis computes 0 ′ . Proof. Combine Dobritsa’s construction with the ACA 0 construction. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 10 / 24

Outline Classical Algebra Background 1 Computing a Basis 2 Computing an Order 3 With A Basis Without A Basis Open Questions 4 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 11 / 24

The Motivating Question Definition An abelian group G = ( G : + , 0 ) equipped with a binary relation ≤ is (totally) ordered if the relation satisfies: antisymmetry (if a ≤ b and b ≤ a , then a = b ), transitivity (if a ≤ b and b ≤ c , then a ≤ c ), totality ( a ≤ b or b ≤ a ), and translation invariance (if a ≤ b , then a + c ≤ b + c ). Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 12 / 24

The Motivating Question Definition An abelian group G = ( G : + , 0 ) equipped with a binary relation ≤ is (totally) ordered if the relation satisfies: antisymmetry (if a ≤ b and b ≤ a , then a = b ), transitivity (if a ≤ b and b ≤ c , then a ≤ c ), totality ( a ≤ b or b ≤ a ), and translation invariance (if a ≤ b , then a + c ≤ b + c ). Theorem (Levi (1942)) An abelian group is orderable if and only if it is torsion-free. Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 12 / 24

The Motivating Question Definition An abelian group G = ( G : + , 0 ) equipped with a binary relation ≤ is (totally) ordered if the relation satisfies: antisymmetry (if a ≤ b and b ≤ a , then a = b ), transitivity (if a ≤ b and b ≤ c , then a ≤ c ), totality ( a ≤ b or b ≤ a ), and translation invariance (if a ≤ b , then a + c ≤ b + c ). Theorem (Levi (1942)) An abelian group is orderable if and only if it is torsion-free. Question Does this remain true in the effective setting? In other words, does every computable torsion-free abelian group admit a computable order? Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 12 / 24

Non-Archimedean Orders on Q κ Example Fixing a basis { b 0 , b 1 } of Q 2 , lexicograph order yields an ordering. Under this order, we have b 0 ≫ b 1 ≫ 0 and so, for example, 1 2 b 0 > 1 2 b 0 − 2 b 1 > b 1 > 0 > − 2 b 0 + 18 b 1 . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 13 / 24

Non-Archimedean Orders on Q κ Example Fixing a basis { b 0 , b 1 } of Q 2 , lexicograph order yields an ordering. Under this order, we have b 0 ≫ b 1 ≫ 0 and so, for example, 1 2 b 0 > 1 2 b 0 − 2 b 1 > b 1 > 0 > − 2 b 0 + 18 b 1 . Example Fixing a basis { b i } i ∈ ω of Q ω , lexicograph order yields an ordering. Under this order, we have b 0 ≫ b 1 ≫ b 2 ≫ · · · ≫ 0 and so, for example, 1 2 b 0 > b 1 + b 2 > b 1 + 2 b 3 > 0 > − b 2 + b 18 > − b 2 . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 13 / 24

Archimedean Orders on Q κ Example Fixing a basis { b 0 , b 1 } of Q 2 and an irrational r ∈ R , the order induced by putting b 0 := 1 ∈ R and b 1 := r is an ordering on Q 2 . √ Thus, for example if r := 2 ≈ 1 . 41, we have 1 . 4 b 0 < b 1 < 1 . 5 b 0 . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 14 / 24

Archimedean Orders on Q κ Example Fixing a basis { b 0 , b 1 } of Q 2 and an irrational r ∈ R , the order induced by putting b 0 := 1 ∈ R and b 1 := r is an ordering on Q 2 . √ Thus, for example if r := 2 ≈ 1 . 41, we have 1 . 4 b 0 < b 1 < 1 . 5 b 0 . Example Fixing a basis { b i } i ∈ ω of Q ω , the order induced by putting b 0 := 1 ∈ R and b i := √ p i for i > 0 is an ordering on Q ω . √ Under this order, we have 1 . 4 b 0 < b 1 < 1 . 5 b 0 (as √ p 1 = 2 ≈ 1 . 41) √ √ and 1 . 2 b 1 < b 2 < 1 . 3 b 1 (as √ p 2 / √ p 1 = 2 ≈ 1 . 22). 3 / Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 14 / 24