Diagram Monoids and Their Congruences

图单群在过去的几年里成为了人们关注的焦点，因为它们在图代数的形成中起着基本的作用，而图代数又在表示理论和理论物理中有应用，也因为它们有趣的代数和组合性质。直到最近，研究主要关注这些一元群的基本半群理论和组合性质，例如格林等价的确定，最小生成集的计算和定义关系，以及幂等函数的表征和计数。PI （NR）和James East （JE）最近开始合作研究这些一元群的另一个重要方面，即它们的同余。在一篇开创性的论文中（与J.D. Mitchell和M.鱼雷合作），他们对有限集合上所有经典图模群上的所有同余进行了分类。在这个过程中，他们开始发展一个看起来很有前途的半群同余的一般理论，至少对有限半群来说是这样。从这项工作中产生了许多方面，每一个方面都很重要，这里提出的项目旨在使这些方面能够得到跟进，并充分发挥其潜力。我们已经确定了四个工作包——处理无限分割monoids，无限图monoids，有限和无限扭曲图monoids，以及一般同余理论的发展——这将在NR对JE的四次研究访问过程中进行调查，为期两年。

Diagram monoids have over the last few years come into sharp focus, because of their fundamental role in the formation of diagram algebras, which in turn have applications in representation theory and theoretical physics, and also because of their intriguing algebraic and combinatorial properties. Until recently, investigations were largely concerned with the basic semigroup-theoretic and combinatorial properties of these monoids, for instance determination of Green's equivalences, computations of minimal generating sets and defining relations, and characterisations and counting of idempotents. PI (NR) and James East (JE) have recently started a collaboration on another important aspect of these monoids, namely their congruences. In a ground-breaking paper (joint with J.D. Mitchell and M. Torpey) they classified all the congruences on all classical diagram monoids over a finite set. In the process they started developing what looks like a promising general theory of semigroup congruences, at least for finite semigroups. Arising from this work are a number of strands, each important in its own right, and the project proposed here is designed to enable these strands to be followed up and for their full potential to be realised. We have identified four work packages -- dealing with infinite partition monoids, infinite diagram monoids, finite and infinite twisted diagram monoids, and development of a general theory of congruences -- which will be investigated in the course of four research visits by NR to JE over a period of two years.