重写系统和Groebner基在计算Kan扩张和关系恒等式中的应用
Applications of Rewriting Systems and Groebner Bases to Computing Kan Extensions and Identities Among Relations
摘要：
本论文集中于重写和Groebner基方法在一系列组合问题中的发展和应用。第二章包含了最重要的结果，这是应用克努特-Benjunt程序Kan扩展，显示如何重写提供了一个有用的方法，试图解决各种组合问题，可以措辞在Kan扩展。第三章证明了标准Knuth-Benzinger算法是逐步的，是Buchberger算法的一个特例。考虑了单侧情况和高维情况。第四章介绍重写系统、Groebner基和自动机。自动机只接受不可约，和自动机的输出约化形式的Kan扩张的介绍进行了讨论。重写系统的归约机用标准的输出自动机来识别，而为代数设计的归约机用Petri网来表示。第五章使用了一个群重写系统的完备化，在算法上确定了一个必要的收缩同伦，以便使用Brown和Razak Salleh设计的覆盖广群方法计算关系之间的恒等式模的生成元集。减少子模块生成器的结果集被确定为Groebner基础问题。算法在GAP 3中实现。
英文摘要：
This thesis concentrates on the development and application of rewriting and Groebner basis methods to a range of combinatorial problems. Chapter Two contains the most important result, which is the application of Knuth-Bendix procedures to Kan extensions, showing how rewriting provides a useful method for attempting to solve a variety of combinatorial problems which can be phrased in terms of Kan extensions. Chapter Three shows that the standard Knuth-Bendix algorithm is step-for-step a special case of Buchberger's algorithm. The one-sided cases and higher dimensions are considered. Chapter Four relates rewrite systems, Groebner bases and automata. Automata which only accept irreducibles, and automata which output reduced forms are discussed for presentations of Kan extensions. Reduction machines for rewrite systems are identified with standard output automata and the reduction machines devised for algebras are expressed as Petri nets. Chapter Five uses the completion of a group rewriting system to algorithmically determine a contracting homotopy necessary in order to compute the set of generators for the module of identities among relations using the covering groupoid methods devised by Brown and Razak Salleh. Reducing the resulting set of submodule generators is identified as a Groebner basis problem. Algorithms are implemented in GAP3.
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