Diagram Monoids and Their Congruences

图幺半群在过去的几年里成为人们关注的焦点，因为它们在图代数的形成中起着重要的作用，而图代数又在表示论和理论物理中有着应用，也因为它们有趣的代数和组合性质。直到最近，调查主要关注的基本半群理论和组合性质的这些monoid，例如确定的绿色的等价，计算最小生成集和定义关系，和特征和计数的幂等元。PI（NR）和James East（JE）最近开始了对这些幺半群的另一个重要方面的合作，即它们的同余。在一篇开创性的论文中（与J.D. Mitchell和M. Torpey），他们分类了有限集合上所有经典图么半群上的所有同余。在这个过程中，他们开始发展什么看起来像一个有前途的一般理论的半群同余，至少对有限半群。从这项工作中产生了若干方面，每一方面都很重要，这里提出的项目旨在使这些方面能够得到贯彻，并充分发挥其潜力。我们已经确定了四个工作包-处理无限分区monoids，无限图monoids，有限和无限扭曲图monoids，和发展的一般理论的同余-这将是调查过程中的四个研究访问NR到乙脑在为期两年。

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.