Abstract
Given a theory T of a polynomially bounded o-minimal expansion R of \({\bar{\mathbb{R}} = \langle\mathbb{R}, +, ., 0, 1, < \rangle}\) with field of exponents \({\mathbb{Q}}\), we introduce a theory \({\mathbb{T}}\) whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that \({\mathbb{T}}\) is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then \({\mathbb{T}}\) axiomatises \({\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}\), where \({\mathbb{R}_{alg}}\) denotes the real algebraic numbers. We describe types and definable sets in our models and prove that \({\mathbb{T}}\) is dependent.
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This paper is based on some results in the author’s Ph.D. thesis titled ‘The first order theory of a dense pair and a discrete multiplicative group’.
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Khani, M. The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two. Arch. Math. Logic 54, 885–898 (2015). https://doi.org/10.1007/s00153-015-0446-7
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DOI: https://doi.org/10.1007/s00153-015-0446-7