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# groupoids

## Calculations with finite groupoids and their homomorphisms

1.71

7 August 2022

Emma J. Moore

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

### Abstract

The groupoids package provides functions for 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. This is described in Section 6.2. It is planned to move this section to a new package Rewriting in time for version 4.11 of GAP.

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

The package was then renamed Gpd and version 1.07 was released in July 2011, ready for GAP 4.5.

Gpd became an accepted GAP package in May 2015.

In April 2017 the package was renamed again, as groupoids.

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

Bug reports, comments, suggestions for additional features, and offers to implement some of these, will all be very welcome.

Please submit any issues at https://github.com/gap-packages/groupoids/issues/ or send an email to the second author at c.d.wensley@bangor.ac.uk.

### Copyright

© 2000-2019, Emma Moore and Chris Wensley.

The groupoids 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 [Hor17] and the package ReleaseTools.

### Contents

1 Introduction
2 Many-object structures
2.1 Magmas with objects; arrows
2.1-1 MagmaWithObjects
2.1-2 IsDomainWithObjects
2.1-3 Arrow
2.1-4 IsSinglePieceDomain
2.2 Semigroups with objects
2.2-1 SemigroupWithObjects
2.3 Monoids with objects
2.3-1 MonoidWithObjects
2.4 Generators of magmas with objects
2.4-1 GeneratorsOfMagmaWithObjects
2.5 Structures with more than one piece
2.5-1 UnionOfPieces
3 Mappings 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
3.3-2 IsInjectiveOnObjects
3.4 Mappings defined by a function
3.4-1 MappingWithObjectsByFunction
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
4.1-6 DirectProductOp
4.2 Groupoid elements; stars; costars; homsets
4.2-1 GroupoidElement
4.2-2 IdentityArrow
4.2-3 Order
4.2-4 ObjectStar
4.3 Subgroupoids
4.3-1 Subgroupoid
4.3-2 SubgroupoidWithRays
4.3-3 SubgroupoidByObjects
4.3-4 SubgroupoidByPieces
4.3-5 FullTrivialSubgroupoid
4.3-6 DiscreteSubgroupoid
4.3-7 SinglePieceSubgroupoidByGenerators
4.4 Left, right and double cosets
4.4-1 RightCoset
4.5 Conjugation
`4.5-1 \^`
4.5-2 ConjugateGroupoid
4.6 Groupoids formed using isomorphisms
4.6-1 GroupoidByIsomorphisms
4.7 Groupoids whose objects form a monoid
4.7-1 SinglePieceGroupoidWithRays
4.7-2 RightActionGroupoid
5 Homomorphisms of Groupoids
5.1 Homomorphisms from a connected groupoid
5.1-1 GroupoidHomomorphismFromSinglePiece
5.2 Properties and attributes of groupoid homomorphisms
5.2-1 Properties of a groupoid homomorphism
5.2-2 Attributes of a groupoid homomorphism
5.2-3 RootGroupHomomorphism
5.2-4 ImagesOfObjects
5.2-5 ImageElementsOfRays
5.2-6 ObjectGroupHomomorphism
5.3 Special types of groupoid homomorphism
5.3-1 InclusionMappingGroupoids
5.3-2 RestrictedMappingGroupoids
5.3-3 IsomorphismNewObjects
5.3-4 IsomorphismStandardGroupoid
5.3-5 IsomorphismPermGroupoid
5.4 Homomorphisms to a connected groupoid
5.4-1 HomomorphismToSinglePiece
5.4-2 GroupoidHomomorphismFromHomogeneousDiscrete
5.5 Homomorphisms to more than one piece
5.5-1 HomomorphismByUnion
5.5-2 IsomorphismGroupoids
5.6 Automorphisms of groupoids
5.6-1 GroupoidAutomorphismByObjectPerm
5.6-2 Automorphisms of a groupoid with rays
5.6-3 AutomorphismGroupOfGroupoid
5.6-4 Inner automorphisms
5.6-5 GroupoidAutomorphismByGroupAutos
5.6-6 AutomorphismGroupoidOfGroupoid
5.7 Matrix representations of groupoids
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 ReducedImageElm
6.4-3 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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