Representation Theory of Semigroups

给定一个集合上的任何数学结构，该集合到其自身的保持结构映射的集合是抽象代数“对象”的一个例子，称为半群。因此，半群在数学中无处不在。另一方面，给定一个抽象定义的半群，什么时候它可以被表示为一个数学结构的映射半群？如果是这样，我们就说它是由动作来表示的。我们的三个机构将进行三个主要的研究方向：行动的自动性（圣安德鲁斯大学），在半群的结构和分类中使用行动和部分行动（约克大学），逆半群的行动（赫瑞瓦特大学）。然而，正如我们在支持案例中所解释的那样，这些线之间存在许多交互。我们的目标是汇集现有的材料，将其放在一个共同的框架中，并利用我们的综合专业知识来解决一些突出的问题。这将以合作的方式完成，与全球该领域的主要研究人员一起完成。使用自动机研究代数，如半群，在代数和理论计算机科学之间建立了一座桥梁，允许我们使用有限状态自动机定义无限代数。自动组和半组现在得到了广泛的研究，但是尽管行为的概念非常依赖，对自动行为的研究（由Dombi引入）还处于起步阶段。自动群的几何结果，如同行者性质的等价性，不适用于半群。我们的目标是利用自动动作来发展自动半群的新概念，这将在一定程度上弥合这些差距。我们将考虑后续的性质，建立新的不可判定结果和算法来计算半群。逆半群是变换伪群的代数形式，是描述几何局部结构的基础。对于每一个逆半群，我们可以关联一个拓扑群，并从这些群中构造C*-代数。因此，逆半群、拓扑群和C*-代数是密切相关的，是非交换几何中的重要组成部分。其指导思想是逆半群的表示理论为研究部分对称性提供了一个统一的框架。这可以看作是对群体表征理论为研究对称性提供统一框架的一种深远的推广。例如，逆半群可以与非周期贴图相关联，而由此产生的群类群是Stone对偶的非交换推广的一部分。此外，已知平铺半群的表示可以控制类群的结构，从而控制相关的C*-代数。一个集合的局部映射（粗略地说，不是处处定义的映射）何时可以（以适当的方式）扩展为全局映射，这是代数和模型理论的核心问题。半群对集合和有序结构的部分作用隐式地应用于许多结构定理中，但尚未得到充分利用。我们将研究半群在具有结构的集合上的部分作用在什么情况下可以“全局”，以及在有限情况下，这个问题是否可判定。我们相信这是解决一些突出问题的关键，例如，是否每个有限逆半群都有有限的f逆覆盖？我们还将利用我们的综合专业知识，试图从经典的行动理论中破解长期未解决的问题。该项目将包括5名长期研究人员：3名提案人，1名研究助理和1名博士生。它还将涉及一系列研究访问和与该领域的主要专家合作。我们将组织一个早期的研讨会，以开始合作过程，并确保我们采取包容性的方法来进行我们的研究。

Given any mathematical structure on a set, the collection of structure-preserving maps of the set to itself is an example of an abstract algebraic `object' called a semigroup. Thus, semigroups pervade mathematics. On the other hand, given an abstractly defined semigroup, when can it be represented as a semigroup of maps of a mathematical structure? If so, we say that it is represented by actions.Three main strands of research will be pursued in our three institutions: automaticity of actions (St. Andrews), the use of actions and partial actions in the structure and classification of semigroups (York), and actions of inverse semigroups (Heriot-Watt). However, as explained in our Case for Support, there are many interactions between these strands. We aim to draw together existing material, place it in a common framework, and use our combined expertise to solve a number of outstanding problems. This will be done in a collaborative way, together with leading researchers in the area from across the globe. Studying algebras such as semigroups using automata builds a bridge between algebra and theoretical computer science, allowing us to define infinite algebras using finite state automata. Automatic groups and semigroups are now widely studied, but although the notion of action is heavily relied upon, the study of automatic actions (introduced by Dombi) is in its infancy. Geometric results for automatic groups, such as the equivalence to the fellow traveller property, do not carry over for semigroups. We aim to use automatic actions to develop new notions of automatic semigroup, which will go some way to bridging these gaps. We will consider subsequent properties, establishing new undecidability results and algorithms to calculate semigroups. Inverse semigroups are the algebraic versions of the pseudogroups of transformations that form the foundation for describing local structures in geometry. With each inverse semigroup one can asssociate an etale topological groupoid and from such groupoids one can construct C*-algebras. Thus inverse semigroups, etale topological groupoids, and C*-algebras are closely related, forming an important ingredient in non-commutative geometry. The guiding idea is that the representation theory of inverse semigroups provides a unifying framework for studying partial symmetries. This can be seen as a far-reaching generalization of the way in which the representation theory of groups provides a unifying framework for studying symmetries. For example, inverse semigroups can be associated with aperiodic tilings, and the groupoids that result form part of a non-commutative generalization of Stone duality. Furthermore, the representations of the tiling semigroups are known to control the structure of the groupoids, and hence the associated C*-algebras.The question of when a partial map of a set (roughly speaking, a map not everywhere defined) can be extended (in a suitable way) to a global map, is central to aspects of algebra and model theory. Partial actions of semigroups on sets and ordered structures are used implicitly in many structure theorems, but yet have not been exploited. We will investigate when the partial action of a semigroup on a set with structure can be `globalised', and, in the finite case, whether this question is decidable. We believe this is the key to solving outstanding questions, such as, does every finite inverse semigroup has a finite F-inverse cover? We will also use our combined expertise to try to crack long unsolved questions from the classical theory of actions.The project will involve 5 permanent researchers: the three proposers, a Research Assistant and a PhD student. It will also involve a string of research visits and collaborations with leading experts in the field. We will organise an early Workshop to begin the collaborative process and to ensure we take an inclusive approach to our research.