Quantifier elimination and decidability of the theory of additive integer group augmented by predicates of multiplicative cyclic submonoids
Wang, Xiaoduo
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https://hdl.handle.net/2142/115816
Description
Title
Quantifier elimination and decidability of the theory of additive integer group augmented by predicates of multiplicative cyclic submonoids
Author(s)
Wang, Xiaoduo
Issue Date
2022-04-26
Director of Research (if dissertation) or Advisor (if thesis)
Hieronymi, Philipp
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Keyword(s)
Model Theory
Logic
Number Theory
Abstract
In this thesis, we study an extension of the theory of the additive group of integers. To be more specific, if we let {q1, q2, ...} be an enumeration of all positive prime integers and for each positive prime integers q, let qN denote the set {qn : n ∈ N}, then we are interested in the theory Th(Z,+,−, 0, 1, qNi )i∈I, where I ⊆ N. We give a set of sentences T explicitly and show that T axiomatizes the theory using a back-and-forth system. We also investigate the extent of quantifier elimination of T and its decidability. In particular, we show that every formula in the language of groups is T-equivalent to a boolean combination of existential formulas, and we also show that the decidability of Th(Z,+,−, 0, 1, qNi )i∈I is equivalent to a number theoretical problem.
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