It is interesting to consider whether a structure can be expanded by an automorphism so that one obtains a nice description of the expanded structure's first-order properties. In this dissertation, we study some such expansions of models of Presburger arithmetic. Building on some of the work of Harnik (1986) and Llewellyn-Jones (2001), in Chapter 2 we use a back-and-forth construction to obtain two automorphisms of sufficiently saturated models of Presburger arithmetic. These constructions are done first in the quotient of the Presburger structure by the integers (which is a divisible ordered abelian group with some added structure), and then lifted to the full Presburger structure.
The first automorphism we construct has special tightly controlled properties that enable us in Chapters 4 and 5 to prove quantifier elimination, decidability, and axiomatizability for both the quotient and the Presburger structure expanded by this automorphism, with explicit axiomatizations given in Chapter 3. The second automorphism is maximal in the sense that its fixed-point set consists only of the standard integers, and has certain properties in common with those of the first automorphism, but we have not attempted to prove quantifier elimination for structures expanded by this automorphism.
In Chapters 6 and 7, we use the quantifier elimination results to describe the definable sets and algebraic closure of the quotient structure and Presburger structure expanded by the first automorphism. This allows us in Chapter 8 to show that the DP-rank in both cases is 2. Finally, in the concluding chapter, we describe some areas of possible future research.
","abstract_html":"<p>It is interesting to consider whether a structure can be expanded by an automorphism so that one obtains a nice description of the expanded structure's first-order properties. In this dissertation, we study some such expansions of models of Presburger arithmetic. Building on some of the work of Harnik (1986) and Llewellyn-Jones (2001), in Chapter 2 we use a back-and-forth construction to obtain two automorphisms of sufficiently saturated models of Presburger arithmetic. These constructions are done first in the quotient of the Presburger structure by the integers (which is a divisible ordered abelian group with some added structure), and then lifted to the full Presburger structure.</p> <p>The first automorphism we construct has special tightly controlled properties that enable us in Chapters 4 and 5 to prove quantifier elimination, decidability, and axiomatizability for both the quotient and the Presburger structure expanded by this automorphism, with explicit axiomatizations given in Chapter 3. The second automorphism is maximal in the sense that its fixed-point set consists only of the standard integers, and has certain properties in common with those of the first automorphism, but we have not attempted to prove quantifier elimination for structures expanded by this automorphism.</p> <p>In Chapters 6 and 7, we use the quantifier elimination results to describe the definable sets and algebraic closure of the quotient structure and Presburger structure expanded by the first automorphism. This allows us in Chapter 8 to show that the DP-rank in both cases is 2. Finally, in the concluding chapter, we describe some areas of possible future research.</p>","abstract_has_math":false,"creators":["Heller, Simon"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Alfred Dolich"],"committee_chairs":[],"committee_members":["Roman Kossak","Phillip Rothmaler","Hans Schoutens"],"year":2019,"date_issued":"2019-09-01T07:00:00Z","date_published":"2019-09-01T07:00:00Z","updated_at":"2026-07-24T01:58:49Z","subjects":["Logic and Foundations","Presburger arithmetic","automorphism","DP-rank","quantifier elimination","expansion"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/3416","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Alfred Dolich"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Roman Kossak","Phillip Rothmaler","Hans Schoutens"]},{"key":"dc:creator","label":"Author","values":["Heller, Simon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-09-05T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Graduate School and University Center of The City University of New York"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Logic and Foundations","Presburger arithmetic","automorphism","DP-rank","quantifier elimination","expansion"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://academicworks.cuny.edu/gc_etds/3416"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It is interesting to consider whether a structure can be expanded by an automorphism so that one obtains a nice description of the expanded structure's first-order properties. In this dissertation, we study some such expansions of models of Presburger arithmetic. Building on some of the work of Harnik (1986) and Llewellyn-Jones (2001), in Chapter 2 we use a back-and-forth construction to obtain two automorphisms of sufficiently saturated models of Presburger arithmetic. These constructions are done first in the quotient of the Presburger structure by the integers (which is a divisible ordered abelian group with some added structure), and then lifted to the full Presburger structure.
The first automorphism we construct has special tightly controlled properties that enable us in Chapters 4 and 5 to prove quantifier elimination, decidability, and axiomatizability for both the quotient and the Presburger structure expanded by this automorphism, with explicit axiomatizations given in Chapter 3. The second automorphism is maximal in the sense that its fixed-point set consists only of the standard integers, and has certain properties in common with those of the first automorphism, but we have not attempted to prove quantifier elimination for structures expanded by this automorphism.
In Chapters 6 and 7, we use the quantifier elimination results to describe the definable sets and algebraic closure of the quotient structure and Presburger structure expanded by the first automorphism. This allows us in Chapter 8 to show that the DP-rank in both cases is 2. Finally, in the concluding chapter, we describe some areas of possible future research.
"]},{"key":"dc:title","label":"Title","values":["Modest Automorphisms of Presburger Arithmetic"]}]}],"canonical_facts":{"dc:contributor.advisor":["Alfred Dolich"],"dc:contributor.committeemember":["Roman Kossak","Phillip Rothmaler","Hans Schoutens"],"dc:creator":["Heller, Simon"],"dc:date.available":["2019-09-05T07:00:00Z"],"dc:description.abstract":["It is interesting to consider whether a structure can be expanded by an automorphism so that one obtains a nice description of the expanded structure's first-order properties. In this dissertation, we study some such expansions of models of Presburger arithmetic. Building on some of the work of Harnik (1986) and Llewellyn-Jones (2001), in Chapter 2 we use a back-and-forth construction to obtain two automorphisms of sufficiently saturated models of Presburger arithmetic. These constructions are done first in the quotient of the Presburger structure by the integers (which is a divisible ordered abelian group with some added structure), and then lifted to the full Presburger structure.
The first automorphism we construct has special tightly controlled properties that enable us in Chapters 4 and 5 to prove quantifier elimination, decidability, and axiomatizability for both the quotient and the Presburger structure expanded by this automorphism, with explicit axiomatizations given in Chapter 3. The second automorphism is maximal in the sense that its fixed-point set consists only of the standard integers, and has certain properties in common with those of the first automorphism, but we have not attempted to prove quantifier elimination for structures expanded by this automorphism.
In Chapters 6 and 7, we use the quantifier elimination results to describe the definable sets and algebraic closure of the quotient structure and Presburger structure expanded by the first automorphism. This allows us in Chapter 8 to show that the DP-rank in both cases is 2. Finally, in the concluding chapter, we describe some areas of possible future research.
"],"dc:identifier":["https://academicworks.cuny.edu/gc_etds/3416"],"dc:subject":["Logic and Foundations","Presburger arithmetic","automorphism","DP-rank","quantifier elimination","expansion"],"dc:title":["Modest Automorphisms of Presburger Arithmetic"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["The Graduate School and University Center of The City University of New York"]},"updated_at":"2026-07-24T01:58:49Z"}