2018/04/03 by Danul K. Gunatilleka, Gunatilleka, Danul K.
Computer Science · Mathematics · #03C65 (Primary) 03C45 (Secondary) #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1804.00932
openalex publication_date 2018/04/03 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
We continue the study of the theories of Baldwin-Shi hypergraphs from [5].\nRestricting our attention to when the rank \δ is rational valued, we show\nthat each countable model of the theory of a given Baldwin-Shi hypergraph is\nisomorphic to a generic structure built from some suitable subclass of the\noriginal class of finite structures with the inherited notion of strong\nsubstructure. We introduce a notion of dimension for a model and show that\nthere is a an elementary chain mathfrakM\β:\β<\ω+1 of\ncountable models of the theory of a fixed Baldwin-Shi hypergraph with\n mathfrakM\β preccurlyeq mathfrakM_\γ if and only if the\ndimension of mathfrakM_\β is at most the dimension of\n mathfrakM_\γ and that each countable model is isomorphic to some\n mathfrakM_\β. We also study the regular types that appear in these\ntheories and show that the dimension of a model is determined by a particular\nregular type. Further, drawing on the work of Brody and Laskowski, we use these\nstructures to give an example of a pseudofinite, \ω-stable theory with a\nnon-locally modular regular type, answering a question of Pillay in [9].\n