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# IdRel

## A package for Identities among Relators

Version 2.05

November 2008

Anne Heyworth
Email: anne.heyworth@googlemail.com

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://www.bangor.ac.uk/~mas023/
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 relators 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

© 2005-2008 Anne Heyworth and Chris Wensley

### Acknowledgements

This idrel package is released under the GNU General Public License (GPL). This file is part of idrel, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

idrel is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

idrel is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with idrel; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.

For more details, see http://www.fsf.org/licenses/gpl.html.

This documentation was prepared with the GAPDoc package of Frank L\"ubeck and Max Neunh\"offer.

### 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
5.3 Module Polynomial Operations
5.3-1 AddTermModulePoly
5.4 Identities among relators
5.4-1 IdentityYSequences
5.4-2 RootIdentities
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