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

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.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Wang, Xiaoduo

Contributors dc:contributor

• Hieronymi, Philipp

• Model Theory
• Logic
• Number Theory

Statement dc:rights

• Copyright 2022 Xiaoduo Wang

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115816

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115816