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LATTICE EMBEDDINGS AND PUNCTUAL LINEAR ORDERS
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 01 August 2025, pp. 1-29
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- Article
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We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers
$\mathbb {Z}$, and the order of the rationals 
$\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order 
$\mathbb {Z}$, we further extend the result to embed the atomless Boolean algebra; we leave the case of 
$\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.
ASSIGNING AN ISOMORPHISM TYPE TO A HYPERDEGREE
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 10 December 2019, pp. 325-337
- Print publication:
- March 2020
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Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let
$f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all 
$x,y \in {2^\omega }$, if 
$x{ \equiv _h}y$ then 
$f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is 
$\omega _1^{CK} + 1$. (2) For all 
$x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.
