Advanced search options

Advanced Search Options 🞨

Browse by author name (“Author name starts with…”).

Find ETDs with:

in
/  
in
/  
in
/  
in

Written in Published in Earliest date Latest date

Sorted by

Results per page:

Sorted by: relevance · author · university · dateNew search

You searched for subject:(Abstract Elementary Classes). Showing records 1 – 2 of 2 total matches.

Search Limiters

Last 2 Years | English Only

No search limiters apply to these results.

▼ Search Limiters

1. Lieberman, Michael Joseph. Topological and Category-Theoretic Aspects of Abstract Elementary Classes.

Degree: PhD, Mathematics, 2009, University of Michigan

We consider the behavior of Galois types in abstract elementary classes (AECs), and introduce several new techniques for use in the analysis of the associated stability spectra. More broadly, we develop novel perspectives on AECs – topological and category-theoretic – from which these techniques flow, and which hold considerable promise as lines of future investigation. After a presentation of the preliminaries in Chapter 2, we give a method of topologizing sets of Galois types over structures in AECs with amalgamation. The resulting spaces – analogues of the Stone spaces of syntactic types – support, among other things, natural correspondences between their topological properties and semantic properties of the AEC (tameness, for example, emerges as a separation principle). In Chapter 4, we note that the newfound topological structure yields a family of Morley-like ranks, along with a new notion of total transcendence. We show that in tame AECs, total transcendence follows from stability in certain cardinals, and that total transcendence, in turn, allows us to bound the number of types over large models. This leads to several upward stability transfer results, one of which generalizes a result of Baldwin, Kueker and VanDieren. The same analysis works in weakly tame AECs provided that they are also weakly stable, a notion that arises in the context of accessible categories. In Chapter 5, we analyze the category-theoretic structure of AECs, and give an axiomatization of AECs as accessible subcategories of their ambient categories of structures. We also give a dictionary for translating notions from the theory of accessible categories into the language of AECs, and vice versa. Weak stability occurs in any accessible category – hence in any AEC – and, since this is what we require to conclude stability in weakly tame AECs, we get the beginnings of a stability spectrum in this context. We close with a curious result: an equivalence between the class of large structures in a categorical AEC and a category of sets with actions of the monoid of endomorphisms of the categoricity structure, effectively reducing the AEC to a simple concrete category. Advisors/Committee Members: Blass, Andreas R. (committee member), Dorais, Francois Gilbert (committee member), Hinman, Peter G. (committee member), Mummert, Carl Beckhorn (committee member), Tappenden, James P. (committee member).

Subjects/Keywords: Model Theory; Nonelementary Classes; Abstract Elementary Classes; Mathematical Logic; Accessible Categories; Category Theory; Mathematics; Science

…circulation, abstract elementary classes—AECs—seem to 3 exhibit the best balance of generality… …abstract elementary classes, the associated notion of Galois type, and the host of properties… …that we retain in passing to abstract elementary classes from classes of structures born of… …that interests us most: abstract elementary classes. First, Definition 2.26. A category C is… …CHAPTER 1 Introduction Classical model theory is the study of elementary classes… 

Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Lieberman, M. J. (2009). Topological and Category-Theoretic Aspects of Abstract Elementary Classes. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/63854

Chicago Manual of Style (16th Edition):

Lieberman, Michael Joseph. “Topological and Category-Theoretic Aspects of Abstract Elementary Classes.” 2009. Doctoral Dissertation, University of Michigan. Accessed July 10, 2020. http://hdl.handle.net/2027.42/63854.

MLA Handbook (7th Edition):

Lieberman, Michael Joseph. “Topological and Category-Theoretic Aspects of Abstract Elementary Classes.” 2009. Web. 10 Jul 2020.

Vancouver:

Lieberman MJ. Topological and Category-Theoretic Aspects of Abstract Elementary Classes. [Internet] [Doctoral dissertation]. University of Michigan; 2009. [cited 2020 Jul 10]. Available from: http://hdl.handle.net/2027.42/63854.

Council of Science Editors:

Lieberman MJ. Topological and Category-Theoretic Aspects of Abstract Elementary Classes. [Doctoral Dissertation]. University of Michigan; 2009. Available from: http://hdl.handle.net/2027.42/63854

2. Drueck, Fred R. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.

Degree: 2013, University of Illinois – Chicago

This dissertation examines three main topics, the topic of defining "superstability" for abstract elementary classes (AECs), uniqueness of limit models, and two cardinal models in abstract elementary classes. In particular we further generalize an analogue of Vaught's theorem which constructs an uncountable two cardinal model starting from the existence of a countable Vaughtian pair in an elementary class to the AEC context originally published by Lessmann, who in turn built upon the work of Grossberg and VanDieren, Shelah, and others. We also give various sufficient conditions on countable models, as well as a condition on models of size kappa that, assuming that a simplified morass,ñ allows us to construct a (kappa^++,kappa)-model. We discuss how this work in AECs to some degree parallels the proof of Jensen's Gap-2 transfer theorem for elementary classes. We also discuss difficulties inherent in proving a true gap-2 transfer theorem for AECs. Additionally, we discuss, progress that has been made toward proving the uniqueness of limit models assuming various "superstability-like" assumptions (much of the work described is due to Shelah, Villaveces, Grossberg, and VanDieren). One small original result is contributed to this discussion. Advisors/Committee Members: Baldwin, John T. (advisor), Marker, David (committee member), Takloo-Bighash, Ramin (committee member), VanDieren, Monica (committee member), Scow, Lynn (committee member).

Subjects/Keywords: Limit Models; Superlimit Models; Two Cardinal Problems; two cardinal models; two cardinal; 2 cardinal; 2 cardinal problems; 2 cardinal model; gap-2 transfer; gap-2; Abstract Elementary Classes; mathematical logic; uniqueness of limit models; morasses; lessmann

…potentially define “superstability” for abstract elementary classes. In particular, we examine… …Theorem for abstract elementary classes. We provide a sufficient condition for the construction… …viii CHAPTER 1 INTRODUCTION Abstract elementary classes (AECs) were introduced… …available in this more general context. In a certain sense, abstract elementary classes strip away… …existence and or transfer of two cardinal models in abstract elementary classes. In classical… 

Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Drueck, F. R. (2013). Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/9996

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Drueck, Fred R. “Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.” 2013. Thesis, University of Illinois – Chicago. Accessed July 10, 2020. http://hdl.handle.net/10027/9996.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Drueck, Fred R. “Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes.” 2013. Web. 10 Jul 2020.

Vancouver:

Drueck FR. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. [Internet] [Thesis]. University of Illinois – Chicago; 2013. [cited 2020 Jul 10]. Available from: http://hdl.handle.net/10027/9996.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Drueck FR. Limit Models, Superlimit Models, and Two Cardinal Problems in Abstract Elementary Classes. [Thesis]. University of Illinois – Chicago; 2013. Available from: http://hdl.handle.net/10027/9996

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.