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A NEW DP-MINIMAL EXPANSION OF THE INTEGERS
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- Journal:
- The Journal of Symbolic Logic / Volume 84 / Issue 2 / June 2019
- Published online by Cambridge University Press:
- 04 March 2019, pp. 632-663
- Print publication:
- June 2019
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We consider the structure
$({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where 
$x|_p y$ means 
$v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe 
${\Bbb Z}$ which is an expansion of 
$({\Bbb Z}, + ,0)$ and a reduct of 
$({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about 
$({\Bbb Z}, + ,0, < )$.
THERE ARE NO INTERMEDIATE STRUCTURES BETWEEN THE GROUP OF INTEGERS AND PRESBURGER ARITHMETIC
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 1 / March 2018
- Published online by Cambridge University Press:
- 01 May 2018, pp. 187-207
- Print publication:
- March 2018
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We show that if a first-order structure
${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then 
${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).
DP-MINIMAL VALUED FIELDS
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 1 / March 2017
- Published online by Cambridge University Press:
- 21 March 2017, pp. 151-165
- Print publication:
- March 2017
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