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# Gpd

## Groupoids, graphs of groups, and graphs of groupoids

Version 1.05

November 2008

Emma Moore
Email: emmajmoore@yahoo.co.uk

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 Gpd package for GAP4 provides functions for the computation with groupoids (categories with every arrow invertible) and their morphisms; for graphs of groups, and graphs of groupoids.

It provides normal forms for Free Products with Amalgamation and for HNN-extensions when the initial groups have rewrite systems and the subgroups have finite index.

The Gpd package was originally implemented in 2000 (as GraphGpd) when the first author was studying for a Ph.D. in Bangor.

The current version is 1.05, released on 21st November 2008, and now includes the more basic structure of magma with objects. It had to be released hurriedly, due to the change of website, so some of the function are no longer available. A new version will be released as soon as possible.

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

### Copyright

© 2000-2008 Emma Moore and Chris Wensley

### Acknowledgements

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

gpd 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.

gpd 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 gpd; 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 Many-object structures
2.1 Magmas with objects
2.1-1 MagmaWithObjects
2.1-2 MultiplicativeElementWithObjects
2.1-3 IsSinglePiece
3 Homomorphisms of many-object structures
3.1 Homomorphisms of magmas with objects
3.1-1 MagmaWithObjectsHomomorphism
4 Groupoids
4.1 Groupoids: their elements and attributes
4.1-1 SinglePieceGroupoid
4.1-2 Pieces
4.1-3 IsPermGroupoid
4.1-4 GroupoidElement
4.1-5 IsSinglePiece
4.2 Subgroupoids
4.2-1 SubgroupoidByPieces
4.3 Stars, Costars and Homsets
4.3-1 ObjectStar
4.3-2 IdentityElement
4.4 Left, right and double cosets
4.4-1 RightCoset
4.5 Conjugation
4.5-1 \^
5 Homomorphisms of Groupoids
5.1 Homomorphisms to a connected groupoid
5.1-1 GroupoidHomomorphism
6 Graphs of Groups and Groupoids
6.1 Digraphs
6.1-1 FpWeightedDigraph
6.2 Graphs of Groups
6.2-1 GraphOfGroups
6.2-2 IsGraphOfFpGroups
6.2-3 RightTransversalsOfGraphOfGroups
6.3 Words in a Graph of Groups and their normal forms
6.3-1 GraphOfGroupsWord
6.3-2 ReducedGraphOfGroupsWord
6.4 Free products with amalgamation and HNN extensions
6.4-1 FreeProductWithAmalgamation
6.4-2 HnnExtension
6.5 GraphsOfGroupoids and their Words
6.5-1 GraphOfGroupoids
6.5-2 GraphOfGroupoidsWord
7 Development History
7.1 Versions of the Package
7.2 What needs to be done next?
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