### Mathematics Preprints 2005

# Computational Discrete Algebra

#### 05.07 : BROWN, R., GHANI, N., HEYWORTH, A. & WENSLEY, C.D.

### String rewriting for double coset systems

(revision of preprint**04.04**)

#### Summary:

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.

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#### Published in:

*J. Symbolic Comp.*41 (2006) 573-590.

#### 05.18 : EVANS, G.A.

### Noncommutative Involutive Bases

#### Summary:

The theory of Gröbner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Gröbner Basis, and indeed in finding alternative methods of computing Gröbner Bases. Two of these methods include the Gröbner Walk method and the computation of Involutive Bases.By the mid 1980's, Buchberger's work had been generalised for noncommutative polynomial rings by Bergman and Mora. This thesis provides the corresponding generalisation for Involutive Bases and (to a lesser extent) the Gröbner Walk, with the main results being as follows.

- Algorithms for several new noncommutative involutive divisions are given, including strong; weak; global and local divisions.
- An algorithm for computing a noncommutative Involutive Basis is given. When used with one of the aforementioned involutive divisions, it is shown that this algorithm returns a noncommutative Gröbner Basis on termination.
- An algorithm for a noncommutative Gröbner Walk is given, in the case of conversion between two harmonious monomial orderings. It is shown that this algorithm generalises to give an algorithm for performing a noncommutative Involutive Walk, again in the case of conversion between two harmonious monomial orderings.
- Two new properties of commutative involutive divisions are introduced (stability and extendibility), respectively ensuring the termination of the Involutive Basis algorithm and the applicability (under certain conditions) of homogeneous methods of computing Involutive Bases.

#### Published in:

University of Wales, Bangor, PhD thesis (September 2005)#### Download:

- evans.pdf

(This pdf file, created from LaTeX source, uses the hyperref package to provide document navigation tools such as bookmarks and clickable links for various items including citations, page numbers and algorithms.) - arXiv : math.RA/0602140