Study on generic structures in model theory

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400203
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kobe University
• Principal Investigator: Kikyo Hirotaka 神戸大学, システム情報学研究科, 教授 (80204824)
• Co-Investigator(Renkei-kenkyūsha): IKEDA Koichiro 法政大学, 経営学部, 教授 (60332029) ; TSUBOI Akito 筑波大学, 大学院数理物質科学研究科, 教授 (30180045)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: ジェネリック構造 / 融合 / モデル完全 / 全融合性 / 任意存在形 / 射影平面

Outline of Final Research Achievements

Consider graphs or hypergraphs as structures. We define a dimension of a structure by the number of points - α× the number of edges. With this dimension, we can define a closed substructure. Given a class of finite structures with some property, we can construct a structure called a generic structure by gluing together these structures. A generic structure reflects closed structure relations between finite structures in the given class. There is a class denoted Kf defined with a boundary function f. If α is a rational number, the generic structure will be model complete under some assumption on f. We have some important lemmas in case that α is irrational. There is a notion of the full amalgamation property. If a class has the full amalgamation property then the generic structure can be axiomatised by universal existential sentences. We constructed an infinite projective plane such that it has no finite projective plane as a substructure.

Free Research Field

数理論理学