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# idrel

## Identities among relations

2.44

4 June 2022

Anne Heyworth

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: https://github.com/cdwensley
Address:
Dr. C.D. Wensley
School of Computer Science
Bangor University
Dean Street
Bangor
Gwynedd LL57 1UT
UK

### Abstract

IdRel is a GAP package 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 last author at c.d.wensley@bangor.ac.uk or submit an issue at the GitHub repository https://github.com/gap-packages/idrel/issues/.

### Copyright

© 1999-2022 Anne Heyworth and Chris Wensley

The IdRel package 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 using the GAPDoc [LN17] and AutoDoc [GH17] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor14] and the package ReleaseTools.

### Contents

1 Introduction
1.1 An illustrative example
2 Rewriting Systems
2.1 Monoid Presentations of FpGroups
2.1-1 FreeRelatorGroup
2.1-2 MonoidPresentationFpGroup
2.1-3 PrintLnUsingLabels
2.1-4 InitialRulesOfPresentation
2.2 Rewriting systems for FpGroups
2.2-1 RewritingSystemFpGroup
2.2-2 OnePassReduceWord
2.2-3 OnePassKB
2.2-4 RewriteReduce
2.2-5 KnuthBendix
2.3 Enumerating elements
2.3-1 ElementsOfMonoidPresentation
3 Logged Rewriting Systems
3.1 Logged Knuth-Bendix Completion
3.1-1 InitialLoggedRulesOfPresentation
3.1-2 LoggedOnePassKB
3.1-3 LoggedRewriteReduce
3.1-4 LoggedKnuthBendix
3.1-5 LoggedRewritingSystemFpGroup
3.2 Logged reduction of a word
3.2-1 LoggedReduceWordKB
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
5.1-2 PrintLnModulePoly
5.2 Components of a module polynomial
5.2-1 Terms
5.3 Module Polynomial Operations
5.3-1 AddTermModulePoly
6 Identities Among Relators
6.1 Constructing identities
6.1-1 RootIdentities
6.1-2 IdentityRelatorSequences
6.1-3 LogSequenceLessThan
6.1-4 ExpandLogSequence
6.2 Identities for S_3
6.2-1 ReduceLogSequences
6.2-2 ConjugateByWordLogSequence
6.2-3 ChangeStartLogSequence
6.2-4 InverseLogSequence
6.2-5 CancelImmediateInversesLogSequence
6.3 Reducing identities
6.3-1 LogSequenceRewriteRules
6.3-2 OnePassReduceLogSequence
6.3-3 MoveRightLogSequence
6.3-4 SubstituteLogSubsequence
6.4 The original approach
6.4-1 IdentitiesAmongRelators
6.4-2 IdentityYSequences
6.5 Partial lists of elements
6.5-1 PartialElementsOfMonoidRepresentation
References
Index
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