Right Noetherian and coherent monoids

我们的建议考虑有限性条件幺半群。一个半群是一个集合加上一个结合二元运算;一个么半群是一个拥有一个单位元的半群。幺半群是最基本的数学结构之一，因为它们代表了组合下集合的自映射（以及更一般的部分映射和关系）的形式框架。实际上，结合性（几乎每一种代数结构都以某种方式享有的性质）和映射的合成（数学的基本运算）是齐头并进的：每一个幺半群M都嵌入到M的自映射幺半群中。由于幺半群中的元素不一定是可逆的，幺半群提供了正确的范式来研究不一定可逆的过程和操作。幺半群在数学中的另一个重要表现是通过单词和连接（自由幺半群），这开辟了与算法，信息和数据处理理论的重要联系。许多类中的有限代数具有使它们易于处理的性质，这是它们有限性的直接结果。我们的建议是出于一种方法，倡导的阿廷和诺特在上个世纪的早期，研究代数满足有限性条件，并已产生了巨大的影响代数的发展。一类代数的有限性条件是所有有限的代数都满足的条件。其思想是，任何代数类中满足给定的条件将有相应的属性所拥有的有限成员，从而产生一个更好的知识，其行为。例如，如果M是有限幺半群，则每个元素都有幂等的幂（元素e使得ee=e）;也就是说，M是周期的。因此，周期性是一个有限性条件，并且是有用的，因为任何周期幺半群，有限或无限，都有一个行为良好的理想结构。鉴于它们与映射的本质联系，幺半群自然作用于集合。通过代数的作用来研究代数是数学的核心。然而，与其他类型的作用（例如作用于向量空间的环或作用于集合的群）不同，为了理解幺半群的作用，我们需要一个称为右同余的某些相容关系的理论。后者是我们进入有限性条件的途径。我们提出了一个雄心勃勃的建议，首先开发数学机器，然后用它来解决一些长期存在的开放问题的幺半群，最后将我们的研究应用到同源领域。鉴于我们的建议的广度，我们将工作分为五个相互关联的主题。这些都是精心建造的，以提供通过技术困难的途径，与意外的应变。主题1发展了一个右同生理论。有了这个工具箱，在主题2和3中，我们研究了右诺特和右连贯的核心有限性条件。第一个要求每一个右同余都是双生成的;对于这样一个基本概念，值得注意的是，主要的问题仍然是开放的-例如，右诺特是否意味着幺半群本身是双生成的。我们建议回答这样的问题，沿着建立一个理解如何与代数结构的这种属性相互作用。右一致性是一个相对概念，因为它保证某些属性传递给子结构。它产生于许多方向，再次，我们建议回答关键的开放性问题，如是否幺半群的所有地图的（无限）集是正确的连贯性。主题4研究了相关的有限性条件，这些条件要么是通过用右理想替换右同余而产生的（就像在其他一些代数结构中会发生的那样），要么是从其他一些数学领域引起我们注意的。最后，在主题5中，我们寻求将我们的结果应用于这些领域。

Our proposal considers finiteness conditions for monoids. A semigroup is a set together with an associative binary operation; a monoid is a semigroup that possesses an identity. Monoids are one of the most fundamental mathematical structures, because they represent a formal framework for self-maps of sets (and more generally partial maps and relations) under composition. Indeed, associativity (a property enjoyed in some way by almost every algebraic structure) and composition of maps (the fundamental operation of mathematics) go hand in hand: every monoid M embeds into the monoid of self-maps of M. Since elements in monoids do not have to be invertible, monoids provide the correct paradigm to study processes and operations that cannot necessarily be reversed. Another important manifestation of monoids in mathematics is via words and concatenation (free monoids), which opens up important links with the theory of algorithms, information and data processing. Finite algebras in many classes possess properties that make them tractable, as a direct consequence of their very finiteness. Our proposal is motivated by an approach, championed by Artin and Noether in the early part of the last century, that studies algebras satisfying finiteness conditions, and which has had an enormous influence on the development of algebra. A finiteness condition for a class of algebras is one that is satisfied by all those that are finite. The idea is that any algebra in the class satisfying the given condition will have properties corresponding to those possessed by the finite members, thus yielding a better knowledge of its behaviour. For example, if M is a finite monoid then every element has a power that is idempotent (an element e such that ee=e); that is, M is periodic. So, periodicity is a finiteness condition, and is useful since any periodic monoid, finite or infinite, has a well-behaved ideal structure.Given their essential connection with maps, monoids naturally act on sets. Studying algebras via their actions is core in mathematics. However, unlike the case for other kinds of actions (such as rings acting on vector spaces or groups acting on sets), to understand actions of monoids, we need a theory of certain compatible relations called right congruences. The latter are our route into finiteness conditions. We present an ambitious proposal to first develop mathematical machinery, then use it to solve a number of long-standing open questions for monoids, and finally apply our research to cognate areas. Given the breadth of our proposal we split the work into five, inter-related, themes. These are carefully constructed to provide pathways through technical difficulties, with contingency for the unexpected. Theme 1 develops a theory of right conguences. Armed with this toolbox, in Themes 2 and 3 we investigate the core finiteness conditions of being right Noetherian and right coherent. The first requires that every right congruence be finitely generated; for such an essential concept it is remarkable that major questions remain open - for instance whether being right Noetherian implies the monoid itself is finitely generated. We propose to answer such questions, along with building an understanding of how this property interacts with algebraic constructions. Right coherency is a relative notion, in the sense that it guarantees certain properties pass to substructures. It arises from many directions and, again, we propose to answer key open questions, such as whether the monoid of all maps of an (infinite) set is right coherent. Theme 4 investigates related finiteness conditions that arise either by replacing right congruences with right ideals (as would happen in some other algebraic structures), or have come to our attention from a number of other areas of mathematics. Finally, in Theme 5, we seek applications of our results to those areas.