Self-similarity and the Universal Steiner Triple System

自相似性和通用斯坦纳三重系统

• 批准号: EP/F017480/1
• 负责人: Bridget Webb
• 金额: $ 1.02万
• 依托单位: The Open University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FF017480%2F1
• 关键词: Self; similarity; Universal; Steiner; Triple

In this project we will look at a particularly interesting infinite structure. The system has proper subsystems which are identical (isomorphic) to the original system. So, inside the structure, there is a copy of itself. Furthermore it contains infinitely many of these 'copies' which are non-nested (they are in fact disjoint except for one triple). Even more interestingly when you put two of these 'copies' together inside the original structure, we make a new structure that is intuitively is more complex than the original (not isomorphic to the original). The structure seems somehow more complex than itself which seems like a paradox .... but it isn't a paradox in the infinite world. This is what makes the infinite world so fascinating.The project uses ideas from Model Theory (a branch of mathematical logic) to answer questions relating to Design Theory. Model Theory is concerned with what you can say about objects in a formal language. We usually use a language which is mathematically expressive enough to capture the essential points about objects in the language but that is as simple as possible. With the tools of Model Theory we will answer questions of self-similarity for this structure. The structure is a countably infinite Steiner triple system. Given a set, X, of v elements (v is greater than or equal to 3) together with a set B of 3-subset (triples) of X such that every 2-subset of X occurs in exactly one triple of B. Then B is called a Steiner triple system.There is much mathematics concerning finite Steiner triple systems. Interesting anomalous properties start to occur for the infinite cases, and it is these that we will investigate.

在这个项目中，我们将研究一个特别有趣的无限结构。该系统具有与原系统相同（同构）的适当子系统。所以，在结构内部，有一个自身的副本。此外，它包含了无限多个非嵌套的“副本”（除了一个三元组之外，它们实际上是不相交的）。更有趣的是，当你把两个这样的“副本”放在原来的结构中时，我们会产生一个新的结构，它直观上比原来的结构更复杂（与原来的结构不同构）。结构似乎比它本身更复杂，这似乎是一个悖论。但在无限的世界里这不是悖论这就是无限世界如此迷人的原因。该项目使用模型论（数学逻辑的一个分支）的思想来回答与设计理论相关的问题。模型论关注的是你可以用形式语言对对象说些什么。我们通常使用一种语言，它在数学上有足够的表达力，可以捕捉到语言中对象的基本点，但这是尽可能简单的。用模型论的工具，我们将回答这个结构的自相似性问题。该结构是一个可数无限Steiner三元系。给定一个v个元素的集合X（v大于或等于3）和一个X的3-子集（三元组）的集合B，使得X的每个2-子集恰好出现在B的一个三元组中。那么B称为Steiner三元系。有趣的反常性质开始出现的无限的情况下，这是我们将调查。