Congruences on Direct Products of Semigroups

点击翻译按钮获取中文摘要

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.