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 selfmaps 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 selfmaps 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 wellbehaved 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 longstanding open questions for monoids, and finally apply our research to cognate areas.
Given the breadth of our proposal we split the work into five, interrelated,
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.
