University of Illinois Urbana-Champaign

Quantifier elimination and decidability of the theory of additive integer group augmented by predicates of multiplicative cyclic submonoids

Wang, Xiaoduo

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WANG-THESIS-2022.pdf

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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.
Graduation Semester
2022-05
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/115816
Copyright and License Information
Copyright 2022 Xiaoduo Wang

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