Anne Heyworth

Computational and Geometric Aspects of Modern Algebra: Rewriting as a special case of non-commutative Gröbner basis theory

Rewriting for semigroups is a special case of Groebner basis theory for noncommutative polynomial algebras. The fact is a kind of folklore but is not fully recognised. The aim of this paper is to elucidate this relationship, showing that the noncommutative Buchberger algorithm corresponds step-by-step to the Knuth-Bendix completion procedure.

String rewriting for double coset systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they (i) do not require the preliminary calculation of cosets; and (ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.

A rewriting alternative to Reidemeister-Schreier

One problem in computational group theory is to find a presentation of the subgroup generated by a set of elements of a group. The Reidemeister-Schreier algorithm was developed in the 1930s and gives a solution based upon enumerative techniques. This however means the algorithm can only be applied to finite groups. This paper proposes a rewriting based alternative to the Reidemeister-Schreier algorithm which has the advantage of being applicable to infinite groups.

Computing over K-modules

Abstract Kan extensions over the category of Sets provide a unifying framework for computation of group, monoid and category actions allowing a number of diverse problems to be solved with a generalised form of string rewriting. This paper extends these techniques to K -algebras and K -categories by using Grobner basis techniques to compute Kan extensions over the category of K-modules.