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University of Wales, Bangor - Mathematics Preprints 2004
Computational Discrete Algebra
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04.04 :
BROWN, R., GHANI, N., HEYWORTH, A. & WENSLEY, C.D.
String-rewriting for double coset systems
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.
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. 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 relations between
them can be revealed.
Finally, we briefly discuss how the double coset problem is a special case of
a problem in computational category theory which indicates that the
rewriting methods we describe are applicable to a much wider class of problem.
Revised as:
05.07.
04.14 : EVANS, G.A.
Noncommutative Involutive Bases
Summary:
Groebner Basis theory originated in the work of Buchberger
and is now considered to be one of the most important and
useful areas of computer algebra.
In 1993, Zharkov and Blinkov proposed an alternative method
of computing a commutative Groebner Basis,
namely the computation of an Involutive Basis.
In the mid 1980's, Mora showed that Buchberger's work could be
generalised for noncommutative rings.
This article explores the issues surrounding
the corresponding generalisation for Involutive Bases,
and constructs a noncommutative involutive division which,
when used with the noncommutative involutive basis algorithm,
returns a noncommutative Groebner Basis on termination.
Published in:
Proc. 10th Int. Conf. Applications of Computer Algebra,
ACA-2004,
Beaumont, Texas,
Q.-N. Tran, editor, (2004) 49-64.
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