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Anne Heyworth, Department of Mathematics and Computer Science, University of Leicester, LE1 7RH UK; Supported by EPSRC grant GR/R29604/01 Kan: A Categorical Approach to Computer Algebra,
Christopher D. Wensley, Department of Informatics, Mathematics Division, University of Wales, Bangor, LL57 1UT, UK
We present a version of the Knuth-Bendix string rewriting procedures for group computations and apply it to the problem of computing the module of identities among relators. By lifting rewriting into the appropriate higher dimension we provide a methodology which is alternative and complementary to the popular geometric approach of pictures.
Combinatorial group theory is the study of groups which are given by means of presentations; these arise naturally in a wide variety of situations including areas as diverse as knot theory , geometry  and cryptography . One of the fundamental problems in computational group theory is the solution of the word problem for a given presentation. The problem is in general undecidable and consequently a number of different approaches have been developed. Amongst the most successful is string rewriting, in particular Knuth-Bendix completion, which attempts to solve the word problem by trying to generate a confluent and Noetherian rewrite system from the presentation. The advantages of this approach are twofold: i) Knuth-Bendix completion can be successfully applied in a large number of situations and; ii) the concrete nature of string rewriting makes these algorithms relatively easy to implement. Indeed, many computer algebra packages solve word problems in precisely this way [11, 18].
Every presentation has associated with it a CW-complex: a cellular model whose fundamental group is the group given by the presentation. The second homotopy group of the CW-complex is the module of identities among relators.
Rewriting for semigroups is a special case of Gröbner basis theory for noncommutative polynomial algebras. The fact is a kind of folklore but is not fully recognised. So our aim in this paper is to elucidate this relationship. A good introduction to string rewriting is, and a recent introduction to non-commutative Gröbner basis theory is. Similarities between the two critical pair completion methods (Knuth-Bendix and Buchberger's algorithm) have often been pointed out in the commutative case. The connection was first observed in and more closely analysed in and more recently in and. In particular it is well known that the commutative Buchberger algorithm may be applied to presentations of abelian groups to obtain complete rewrite systems.
Rewriting involves a presentation sgp〈X|R〉 of a semigroup S and presents S as a factor semigroup X† / =R where X† is the free semigroup on X and =R is the congruence generated by the subset R of X† × X†. Non-commutative Gröbner basis theory involves a presentation alg〈X|F〉 of a non-commutative algebra A over a field K and presents A as a factor algebra K[X†]/〈F〉 where K[X†] is the free K-algebra on the semigroup X† and 〈F〉 is the ideal generated by F, a subset of K[X†].
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