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# IdRel

## A package for Identities among Relators

Version 2.31

01/06/2015

Anne Heyworth

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://pages.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).

IdRel became an accepted GAP package in May 2015.

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

### Copyright

© 1999-2015 Anne Heyworth and Chris Wensley

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.

### Acknowledgements

This documentation was prepared with the GAPDoc package of Frank Lübeck and Max Neunhöffer.

### 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
References
Index
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