Right Noetherian and coherent monoids

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.

As a team we are always aware that we work in a community of scholars embedded in wider society. We carry an important responsibility to conduct our research so as to have the greatest possible impact on both our academic community and, more broadly, science and society. Our ambitious and wide-ranging project is designed to maximise potential benefits. Here we outline our vision, and give further details in `Pathways to Impact'.

Academic Impact: Our work will impact upon a variety of academic disciplines: naturally, semigroup theory and algebra, and also development of computer algebra packages (specifically, GAP), universal algebra, and potentially some areas of model theory and analysis. St Andrews is a world leader for combinatorial and computational algebra, the host of Centre for Interdisciplinary Research in Computational Algebra (CIRCA), and a centre for management and development of GAP. The project is thus a perfect vehicle for deepening connections between abstract and computational algebra.

UK Prestige: The project will have tangible impact on the prestige of UK mathematics. The representation theory of monoids by their actions on sets is a central concept, yet, once it moves beyond the early routines, throws up significant and fundamental difficulties. Recent breakthroughs by the PIs and others have brought the field to a stage where the possibility of a novel strong theory has opened up, which would enable progress on the remaining difficult questions. Currently the UK is at the forefront of this area: the funded project will play a vital role in keeping us there. It will provide a platform for extended collaborations, both national and international, and thus increase presence and visibility of UK mathematics.

Impact on Training: The project will provide a careful training programme for two RAs. By the end of the project they will be fully equipped to be independent researchers and train the next generation. More broadly, the developing world continues to value the training and rigour that mathematics research brings, and look to nations such as the UK to provide it - let us continue to do so. Both York and St Andrews are flagship teaching institutions with a strong international presence. The momentum and enthusiasm generated by an active major research programme filters down to the undergraduate community via our teaching-research linkages.

Links with European mathematicians: As a consequence of UK leaving the EU, there is a danger that a damaging distance opens up between the UK and European academic institutions. With our array of key collaborators from EU and other European countries this project will make a timely positive contribution towards countering this.

Impact on Sino-British links: China has emerged as a strong academic force, but there are some significant differences in academic practice, and communication difficulties, which may hamper collaboration and progress. Deepening our joint work with Yang Dandan (Xidian), and opening up new links as and when appropriate, will facilitate a valuable bridge between the two academic cultures.

Impact on information/communication technologies: Our work on finitely presented structures and algorithmic decidability has potential for developing encryption systems. We will consider preservation properties under direct products; the next step to semi-direct products has immediate connections with existing cryptographic protocols.

Public Engagement: The team will take the opportunities provided by networks such as the `York Festival of Ideas', `Pint of Science' and St Andrews outreach initiatives to promote their work.

Impact on Gender Equality: There is a woeful shortage of female role models in mathematics, which may discourage young women from entering the field. A dynamic team with a senior female mathematician at its helm, and several other female collaborators, can only be good for the wider mathematical and social culture.