This project is an extention of Project A15. We will focus on the following question: (*) Can one characterize the monoids that admit a finite complete rewriting system by means of a number of invariant properties? The properties FHT of finite homological type and FDT of finite derivation type are examples of invariant properties that these monoids satisfy. There are various possibilities for obtaining analogues of these properties in higher dimensions. This is one line of research we will pursue. It seems probable that a solution to (*) (if there is one) will require some geometric and/or topological input. We therefore regard it as important to investigate the higher dimensional geometric aspects of rewriting systems.
In his thesis D. Cruickshank showed how to associate a chain of ideals in a certain commutative ring with a group (and more generally a monoid) of type FP_n, and he investigated some properties of these chains. Since monoids with finite complete rewriting systems are of type FP_\infty, these chains exist in all dimensions. It may be that there are additional properties of these chains in this case, and we intend to investigate this.
Finally, it may be that the notion of completeness is too rigid for characterization, but that some slightly reformulated concept involving a ``finite amount of flexibility'' could be characterized. In this respect, both geometric and language theoretic concepts would come into play. This is another topic for our research.
Supported by a grant from the Engineering and Physical Sciences Research Councel of the UK under the MathFIT 2000 initiative.