Congruences on Direct Products of Semigroups

本项目的总体目标是对以下基本问题进行系统的研究：用S和T的同余刻画两个半群的直积S和T上的同余关系。同余和直积是一般代数中最重要的基本概念之一：它们都使已知的对象可以构造出新的对象。在群论中，同余与正规子群一一对应，直积的正规子群有一个简洁的刻画：它们是满足一定附加交换子条件的正规子群的次直积。这一描述扩展到其他同余可换簇，如环、结合代数和李代数。然而，即使在群的情况下，刻画具有两个以上因子的直积的正规子群也是一个既难又有趣的问题。半群是另一类被广泛研究的代数结构，在数学和理论计算机科学中都有广泛的应用。半群不是同余置换的，它们的同余不能归结为子半群，这通常使半群理论中的同余研究具有挑战性。尽管如此，从Malcev[M52，M53]的开创性结果到今天，对半群同余的研究一直是半群理论发展的一条不变的线。这个项目的目的是在理解两个半群的直积上的同余方面迈出第一步。我们将从分别研究‘类群’半群[H95]和‘群平凡’半群的直积上的同余开始。接下来的问题是考虑一个‘类群’半群与一个‘群平凡’半群的直积。另一个目标是研究与次直积之间的联系。半群S上的同余是S和S的次直积，但S和T上的同余在什么程度上可以描述为S的同余和T的同余的次直积？解决这个问题将涉及发展一种被视为次要乘积的新的同余理论。对于同余可换变种，它们的特征是包含对角线{(S，S)：SS}，但对于半群没有这样直接的刻划。该项目还可能包括对单边同余的研究。虽然这似乎是一个更难的问题，但情况可能并非如此：群上的单边同余与子群一一对应，而直积的子群只是因子的子群的次直积。这在半群理论中是一个诱人的问题，因为更好地理解直积上的单边同余可能会产生一个长期悬而未决的问题的解决方案，即两个Noether么半群的直积是否一定是Noether的。Noether么半群是所有右同余都有限生成的么半群。这一建议是目前半群代数理论和一般代数研究兴趣的前沿，非常适合圣安德鲁斯的代数和组合学研究小组的研究环境，提供了许多外部合作链接。成功的结果将代表着对代数半群理论的重大贡献，并将为进一步的学术工作开辟一条道路。

The overarching aim of this project is to undertake a systematic investigation into the following fundamental question: Describe the congruence relations on the direct product S and T of two semigroups in terms of congruences of S and T.Congruences and direct products are among the most important fundamental notions in general algebra: they both enable new objects to be constructed from the known ones. In group theory, congruences are in one-one correspondence with normal subgroups, and there is a neat description of normal subgroups of direct products: they are subdirect products of normal subgroups which satisfy a certain additional commutator condition. This description extends to other congruence permutable varieties, such as rings, associative algebras and Lie algebras. However, even in the case of groups, describing the normal subgroups of direct products with more than two factors is a hard and interesting problem.Semigroups are another type of widely studied algebraic structures, and they have a broad range of applications in both mathematics and theoretical computer science. Semigroups are not congruence permutable, and their congruences cannot be reduced to subsemigroups, typically making a study of congruences in semigroup theory challenging. Nonetheless, investigation of congruences of semigroups has been a constant strand in the development of the theory of semigroups, from the seminal results of Malcev [M52, M53] to the present day. The purpose of this project is to undertake the first steps towards understanding the congruences on the direct product of two semigroups. We will begin by separately investigating the congruences on the direct products of 'group-like' semigroups [H95] and 'group-trivial' semigroups. A following line of enquiry is to consider the direct product of a 'group-like' semigroup with a 'group-trivial' semigroup.A further objective is to investigate links with subdirect products. Congruences on a semigroup S are a subdirect products inside S and S, but to what extent can congruences on S and T be described as subdirect products of a congruence of S and a congruence of T? Tackling this question will involve developing a new theory of congruences viewed as subdirect products. For congruence permutable varieties, these are characterised by containing the diagonal {(s,s):sS}, but no such straightforward characterisation is available for semigroups.The project may also include an investigation into one-sided congruences. While this would appear to be an even harder problem, this may not be the case: one-sided congruences on a group are in one-one correspondence with subgroups, and subgroups of a direct product are simply subdirect products of subgroups of the factors. This is an enticing problem in semigroup theory, as a better understanding of one-sided congruences on direct products would likely yield the solution to the long-standing open problem of whether the direct product of two Noetherian monoids is necessarily Noetherian. Noetherian monoids are those in which all right congruences are finitely generated. This proposal is at the cutting edge of current research interest in algebraic theory of semigroups and general algebra and fits very well with the research environment in the Algebra and Combinatorics Research Group in St Andrews, offering numerous external collaborative links. Successful outcome would represent a major contribution to algebraic semigroup theory and would open up a path into further academic work.