Representation Theory of Semigroups

半群表示论

• 批准号: EP/I032312/1
• 负责人: Victoria Gould
• 金额: $ 31.78万
• 依托单位: University of York
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI032312%2F1
• 关键词: Representation; Theory; Semigroups

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.

给定集合上的任何数学结构，集合到自身的结构保持映射的集合是称为半群的抽象代数“对象”的一个例子。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，什么时候它可以表示为一个具有数学结构的映射半群？如果是这样的话，我们说它是由行动所代表的。我们的三个机构将进行三个主要的研究：行动的自动性（圣安德鲁斯），在半群的结构和分类中使用行动和部分行动（约克），以及逆半群的行动（赫瑞瓦特）。然而，正如我们在支持案例中所解释的那样，这些股之间有许多相互作用。我们的目标是将现有的材料汇集在一起，将其置于一个共同的框架中，并利用我们的综合专业知识来解决一些悬而未决的问题。这项工作将与该领域来自地球仪的主要研究人员合作进行。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，使我们能够使用有限状态自动机定义无限代数。自动群和半群现在被广泛地研究，但是尽管动作的概念被严重依赖，自动动作的研究（由Dombi引入）仍处于起步阶段。自动群的几何结果，如同路者性质的等价性，不适用于半群。我们的目标是使用自动行动，发展自动半群的新概念，这将在一定程度上弥合这些差距。我们将考虑随后的性质，建立新的不可判定性结果和算法来计算半群。逆半群是伪变换群的代数形式，伪变换群是描述几何中局部结构的基础。每一个逆半群都可以对应一个标准拓扑广群，从这样的广群可以构造C *-代数。因此，逆半群，etale拓扑群胚和C *-代数是密切相关的，形成了非交换几何的重要组成部分。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。这可以被看作是群的表示论为研究对称性提供了一个统一框架的方式的一个意义深远的概括。例如，逆半群可以与非周期镶嵌相关联，并且由此产生的群胚形成了Stone对偶的非交换推广的一部分。此外，我们知道镶嵌半群的表示控制着群胚的结构，因此也控制着相关的C *-代数。当一个集合的部分映射（粗略地说，一个不是处处定义的映射）可以扩展（以适当的方式）到一个整体映射时，这个问题是代数和模型论的核心。半群在集合和序结构上的部分作用在许多结构定理中被隐含地使用，但尚未被利用。我们将研究半群在具有结构的集合上的部分作用何时可以“全局化”，以及在有限的情况下，这个问题是否是可判定的。我们相信这是解决一些悬而未决的问题的关键，例如，是否每个有限逆半群都有一个有限的F-逆覆盖？我们还将利用我们的综合专业知识，试图破解经典行动理论中长期未解决的问题。该项目将涉及5名永久研究人员：三名提议者，一名研究助理和一名博士生。它还将涉及一系列研究访问和与该领域领先专家的合作。我们将组织一个早期的研讨会，开始合作过程，并确保我们采取包容性的方法，我们的研究。

项目成果

Coherency, free inverse monoids and related free algebras

一致性、自由逆幺半群和相关自由代数

• DOI: 10.1017/s0305004116000505
• 发表时间: 2016
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: GOULD V

Ehresmann monoids

• DOI: 10.1016/j.jalgebra.2015.06.035
• 发表时间: 2014
• 期刊: Cell Death & Disease
• 影响因子: 9
• 作者: Mário J. J. Branco-Mário-J.-J.-Branco-2070290661;Gracinda M. S. Gomes;Victoria Gould

Automatic semigroup acts

自动半群行为

• DOI: 10.1016/j.jalgebra.2015.03.032
• 发表时间: 2015
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Dombi E

Ehresmann monoids: Adequacy and expansions

埃雷斯曼幺半群：充分性和扩展

• DOI: 10.1016/j.jalgebra.2018.06.036
• 发表时间: 2018-11
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Mario J.J. Branco;Gracinda M.S. Gomes;Victoria Gould;Yanhui Wang
• 通讯作者: Yanhui Wang

COHERENCY AND CONSTRUCTIONS FOR MONOIDS

• DOI: 10.1093/qmath/haaa040
• 发表时间: 2020-12-01
• 期刊: QUARTERLY JOURNAL OF MATHEMATICS
• 影响因子: 0.7
• 作者: Dandan, Yang;Gould, Victoria;Zenab, Rida-E
• 通讯作者: Zenab, Rida-E