Right Noetherian and coherent monoids

我们的建议考虑了一元群的有限条件。半群是具有关联二元运算的集合；单群是一个具有恒等的半群。一元群是最基本的数学结构之一，因为它们代表了组合下集合（以及更一般的部分映射和关系）的自映射的形式框架。事实上，结合性（几乎所有代数结构都以某种方式享有的性质）和映射的组成（数学的基本操作）是齐头齐头的：每个一元群M嵌入到M的自映射的一元群中。因为一元群中的元素不一定是可逆的，所以一元群提供了正确的范例来研究那些不一定可逆的过程和操作。一元群在数学中的另一个重要表现是通过词和串接（自由一元群），它开辟了与算法、信息和数据处理理论的重要联系。有限代数在许多类中具有易于处理的性质，这是它们的有限性的直接结果。我们的提议是由Artin和Noether在上世纪早期倡导的一种方法推动的，该方法研究满足有限条件的代数，并对代数的发展产生了巨大的影响。一类代数的有限条件是所有有限代数都满足的有限条件。其思想是，类中满足给定条件的任何代数都具有与有限成员所拥有的属性相对应的属性，从而更好地了解其行为。例如，如果M是有限单群，则每个元素都有幂等的幂次（元素e使得ee=e）；也就是说，M是周期的。因此，周期性是一个有限条件，它是有用的，因为任何周期单群，无论是有限的还是无限的，都有一个表现良好的理想结构。鉴于它们与映射的本质联系，一元群自然作用于集合。通过代数的行为来研究代数是数学的核心。然而，与其他类型的作用（如作用于向量空间的环或作用于集合的群）不同的是，为了理解单群的作用，我们需要一种称为右同余的相容关系的理论。后者是我们进入有限条件的途径。我们提出了一个雄心勃勃的建议，首先发展数学机制，然后用它来解决一些长期存在的monoids开放问题，最后将我们的研究应用于相关领域。考虑到我们提案的广度，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，并为意外情况提供应急。主题1发展了权利结果理论。有了这个工具箱，在主题2和主题3中，我们将研究正确的Noetherian和正确的coherent的核心有限条件。第一个要求每个右同余是有限生成的；对于这样一个重要的概念，值得注意的是，主要的问题仍然是开放的——例如，作为正确的诺etherian是否意味着单群本身是有限生成的。我们建议回答这些问题，以及建立对这种性质如何与代数结构相互作用的理解。右相干是一个相对的概念，从某种意义上说，它保证了某些属性传递给子结构。它来自许多方向，我们再次提出回答关键的开放性问题，例如（无限）集合的所有映射的单阵是否是正确相干的。主题4研究了相关的有限条件，这些条件要么是通过用正确的理想代替正确的同余而产生的（就像在其他一些代数结构中发生的那样），要么是从许多其他数学领域引起我们的注意。最后，在主题5中，我们寻求将我们的成果应用于这些领域。

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.