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# IdRel

## A package for Identities among Relators

Version 2.03

October 2007

Anne Heyworth
e-mail: anne.heyworth@googlemail.com

Chris Wensley
e-mail: c.d.wensley@bangor.ac.uk
WWW: http://www.informatics.bangor.ac.uk/~cwensley/
Address:
School of Computer Science, Bangor University,
Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.

### Abstract

The IdRel package was originally implemented in 1999, using the GAP 3 language, when the first author was studying for a Ph.D. in Bangor.

This package is designed to compute a minimal set of generators for the module of the identities among relations of a group presentation. It does this using

- rewriting and logged rewriting: a self-contained implementation of the Knuth-Bendix process using the monoid presentation associated to the group presentation;

- monoid polynomials: an implementation of the monoid ring;

- module polynomials: an implementation of the right module over this monoid generated by the relators.

- Y-sequences: used as a rewriting way of representing elements of a free crossed module (products of conjugates of group relators and inverse relators).

Bug reports, suggestions and comments are, of course, welcome. Please contact the second author at c.d.wensley@bangor.ac.uk.

### Copyright

© 2005-2007 by Anne Heyworth and Chris Wensley

We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.

### Contents

1. Introduction
2. Rewriting Systems
2.1 Identity Y-sequences
2.2 Monoid Presentations of FpGroups
2.2-1 FreeRelatorGroup
2.2-2 MonoidPresentationFpGroup
2.3 Rewriting systems for FpGroups
2.3-1 RewritingSystemFpGroup
2.3-2 OnePassReduceWord
2.3-3 OnePassKB
2.4 Enumerating elements
2.4-1 ElementsOfMonoidPresentation
3. Logged Rewriting Systems
3.1 Logged Knuth-Bendix Completion
3.1-1 LoggedOnePassKB
3.1-2 LoggedKnuthBendix
3.2 Logged reduction of a word
3.2-1 LoggedReduceWordKB
3.2-2 LoggedRewritingSystemFpGroup
4. Monoid Polynomials
4.1 Construction of monoid polynomials
4.1-1 MonoidPolyFromCoeffsWords
4.2 Components of a polynomial
4.2-1 Terms
4.2-2 Monic
4.2-3 AddTermMonoidPoly
4.3 Monoid Polynomial Operations
4.3-1 Length
4.4 Reduction of a Monoid Polynomial
4.4-1 ReduceMonoidPoly
5. Module Polynomials
5.1 Construction of module polynomials
5.1-1 ModulePoly
5.2 Components of a module polynomial
5.2-1 Terms
5.2-2 ZeroModulePoly
5.2-3 AddTermModulePoly
5.3 Module Polynomial Operations
5.4 Identities among relators
5.4-1 IdentitiesAmongRelators
5.4-2 RootIdentities
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