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# Gpd

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

Version 1.45

02/11/2016

Emma Moore

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 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. The most basic structure introduced is that of magma with objects, followed by semigroup with objects, then monoid with objects and finally groupoid which is a group with objects.

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.

Version 1.07 was released in July 2011, to be tested with GAP 4.5. Version 1.15 came out with the first release of GAP 4.5 in June 2012, and was submitted for official acceptance as a GAP package. Gpd became an accepted GAP package in May 2015. The latest version is 1.45 of 2nd November 2016, prepared for GAP 4.8.

Recent versions implement many of the constructions described in the paper [AW10] for automorphisms of groupoids.

Bug reports, suggestions and comments are, of course, welcome. Please submit an issue at http://github.com/gap-packages/gpd/issues/ or send an email to the last author at c.d.wensley@bangor.ac.uk.

### Copyright

© 2000-2016 Emma Moore and Chris Wensley

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.

### Acknowledgements

This documentation was prepared with the GAPDoc package [LN12] of Frank Lübeck and Max Neunhöffer.

The procedure used to mount new releases on GitHub uses the packages GitHubPagesForGAP [Hor14] and ReleaseTools of Max Horn.

### Contents

1 Introduction
2 Many-object structures
2.1 Magmas with objects; arrows
2.1-1 MagmaWithObjects
2.1-2 Arrow
2.1-3 IsSinglePiece
2.2 Semigroups with objects
2.2-1 SemigroupWithObjects
2.3 Monoids with objects
2.3-1 MonoidWithObjects
2.4 Structures with one or more pieces
2.4-1 DomainWithSingleObject
2.4-2 UnionOfPieces
3 Homomorphisms of many-object structures
3.1 Homomorphisms of magmas with objects
3.1-1 MagmaWithObjectsHomomorphism
3.2 Homomorphisms of semigroups and monoids with objects
3.3 Homomorphisms to more than one piece
3.3-1 HomomorphismByUnion
4 Groupoids
4.1 Groupoids: their properties and attributes
4.1-1 SinglePieceGroupoid
4.1-2 ObjectList
4.1-3 IsPermGroupoid
4.1-4 UnionOfPieces
4.1-5 HomogeneousGroupoid
4.2 Groupoid elements; stars; costars; homsets
4.2-1 Arrow
4.2-2 IdentityArrow
4.2-3 Order
4.2-4 ObjectStar
4.3 Subgroupoids
4.3-1 Subgroupoid
4.3-2 SubgroupoidWithRays
4.4 Left, right and double cosets
4.4-1 RightCoset
4.5 Conjugation
4.5-1 ConjugateArrow
4.5-2 SinglePieceSubgroupoidByGenerators
4.5-3 ConjugateGroupoid
5 Homomorphisms of Groupoids
5.1 Homomorphisms from a connected groupoid
5.1-1 GroupoidHomomorphismFromSinglePiece
5.2 Homomorphisms to a connected groupoid
5.2-1 HomomorphismToSinglePiece
5.3 Homomorphisms to more than one piece
5.3-1 HomomorphismByUnion
5.4 Groupoid automorphisms
5.4-1 GroupoidAutomorphismByObjectPerm
5.4-2 Automorphisms of a groupoid with rays
5.4-3 AutomorphismGroupOfGroupoid
5.4-4 GroupoidAutomorphismByGroupAutos
5.4-5 RootGroupHomomorphism
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 Technical Notes
7.1 Many object structures
7.2 Many object homomorphisms
8 Development History
8.1 Versions of the Package
8.2 What needs to be done next?
References
Index
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