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# Gpd

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

Version 1.03

October 2007

Emma Moore
e-mail: emmajmoore@yahoo.co.uk

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 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 1.03, released on October 8th 2007.

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

### Copyright

© 2000-2007 by Emma Moore and Chris Wensley

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

### Contents

1. Introduction
2. Groupoids
2.1 Groupoids: their elements and attributes
2.1-1 ConnectedGroupoid
2.1-2 ComponentsOfGroupoid
2.1-3 IsPermGroupoid
2.1-4 GroupoidElement
2.1-5 IsConnectedGroupoid
2.2 Subgroupoids
2.2-1 SubgroupoidByComponents
2.3 Stars, Costars and Homsets
2.3-1 ObjectStar
2.3-2 IdentityElement
2.4 Left, right and double cosets
2.4-1 RightCoset
3. Morphisms of Groupoids
3.1 Morphisms to a connected groupoid
3.1-1 MorphismOfConnectedGroupoids
3.1-2 MorphismToConnectedGroupoid
3.1-3 IsomorphismNewObjects
3.1-4 InclusionMappingGroupoids
3.2 General Morphisms and composites
3.2-1 GroupoidMorphismByComponents
3.2-2 IsGroupoidMorphism
3.2-3 ImagesOfObjects
3.2-4 Product
3.2-5 InverseMorphism
4. Graphs of Groups and Groupoids
4.1 Digraphs
4.1-1 FpWeightedDigraph
4.2 Graphs of Groups
4.2-1 GraphOfGroups
4.2-2 IsGraphOfFpGroups
4.2-3 RightTransversalsOfGraphOfGroups
4.3 Words in a Graph of Groups and their normal forms
4.3-1 GraphOfGroupsWord
4.3-2 ReducedGraphOfGroupsWord
4.4 Free products with amalgamation and HNN extensions
4.4-1 FreeProductWithAmalgamation
4.4-2 HnnExtension
4.5 GraphsOfGroupoids and their Words
4.5-1 GraphOfGroupoids
4.5-2 GraphOfGroupoidsWord
5. Development History
5.1 Versions of the Package
5.2 What needs to be done next?
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