Mathematical Sciences: Presentations of Inverse Monoids

这项研究涉及逆么半群的表示理论的发展。将研究的问题大致有两个领域。第一个是关于由生成元和关系表示逆么半群时产生的基本决策问题。特别是，将研究字问题和E-酉性问题，最初的重点是一个关系者的情况。将使用组合群论中的许多几何技术，以及语言理论和自动机理论的基本结果。第二个问题是关于自由逆么半群的闭逆子么半群的研究及相关问题。特别地，我们将详细研究自由逆么半群的闭逆子么半群、有限图的拓扑、自由逆么半群的有理可识别子集和有限非周期逆么半群的点深度之间的关系。此外，还将尝试开始部分1-1变换的有限逆半群的计算理论。这项研究涉及么半群理论，它是上个世纪引入的最简单的抽象代数对象。研究人员感兴趣的正是自动机理论和计算机科学中的形式语言理论中出现的那些结构。他们的技术范围从群论到几何再到计算。他们解决问题的方法富有想象力。这一建议应该对数学和计算机科学做出重要贡献。

This research is concerned with the development of a theory of presentations of inverse monoids. There are two general areas of problems which will be studied. The first is concerned with basic decision problems which arise when an inverse monoid is presented by generators and relations. In particular, the word problem and the E-unitary problem will be studied, with initial emphasis focussed on the one relator case. Many of the geometric techinques from combinatorial group theory as well as basic results from language theory and automata theory will be used. The second problem is concerned with a study of the closed inverse submonoids of the free inverse monoid and related problems. In particular, the relationship between the closed inverse submonoids of a free inverse monoid, the topology of finite graphs, the rational and recognizable subsets of a free inverse monoid and the dot depth of a finite aperiodic inverse monoid will be studied in detail. In addition, an attempt will be made to begin a computational theory of finite inverse semigroups of partial one-one transformations. This research concerns the theory of monoids, the simplest of the abstract algebraic objects introduced in the last century. The particular ones that the investigators are interested in are precisely those structures arising in automata theory and the theory of formal languages in computer science. Their techniques range from group theory to geometry to computations. Their approaches to the problems are imaginative. This proposal should make important contributions to both mathematics and computer science.