  
  
                                  [1X groupoids [101X
  
  
          [1X Calculations with finite groupoids and their homomorphisms [101X
  
  
                                      1.68
  
  
                                4 September 2019
  
  
                                 Emma J. Moore
  
                             Christopher D. Wensley
  
  
  
  Christopher D. Wensley
      Email:    [7Xmailto:c.d.wensley@bangor.ac.uk[107X
      Homepage: [7Xhttp://pages.bangor.ac.uk/~mas023/[107X
      Address:  [33X[0;14YDr. C.D. Wensley[133X
                [33X[0;14YSchool of Computer Science and Electronic Engineering[133X
                [33X[0;14YBangor University[133X
                [33X[0;14YDean Street[133X
                [33X[0;14YBangor[133X
                [33X[0;14YGwynedd LL57 1UT[133X
                [33X[0;14YUK[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5Xgroupoids[105X  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  [13Xmagma with objects[113X, followed by [13Xsemigroup with objects[113X, then [13Xmonoid with
  objects[113X and finally [13Xgroupoid[113X which is a [13Xgroup with objects[113X.[133X
  
  [33X[0;0YIt  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 [14X6.2[114X. It is planned
  to  move this section to a new package [5XRewriting[105X in time for version 4.11 of
  [5XGAP[105X.[133X
  
  [33X[0;0YThe  [5Xgroupoids[105X package was originally implemented in 2000 (as [5XGraphGpd[105X) when
  the first author was studying for a Ph.D. in Bangor.[133X
  
  [33X[0;0YThe package was then renamed [5XGpd[105X and version 1.07 was released in July 2011,
  ready for [5XGAP[105X 4.5.[133X
  
  [33X[0;0Y[5XGpd[105X became an accepted [5XGAP[105X package in May 2015.[133X
  
  [33X[0;0YIn April 2017 the package was renamed again, as [5Xgroupoids[105X.[133X
  
  [33X[0;0YRecent  versions  implement many of the constructions described in the paper
  [AW10] for automorphisms of groupoids.[133X
  
  [33X[0;0YBug  reports,  comments,  suggestions for additional features, and offers to
  implement some of these, will all be very welcome.[133X
  
  [33X[0;0YPlease              submit             any             issues             at
  [7Xhttps://github.com/gap-packages/groupoids/issues/[107X  or  send  an email to the
  second author at [7Xmailto:c.d.wensley@bangor.ac.uk[107X.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2000-2019, Emma Moore and Chris Wensley.[133X
  
  [33X[0;0YThe  [5Xgroupoids[105X  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.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis  documentation  was prepared using the [5XGAPDoc[105X [LN17] and [5XAutoDoc[105X [GH17]
  packages.[133X
  
  [33X[0;0YThe   procedure   used   to   produce   new   releases   uses   the  package
  [5XGitHubPagesForGAP[105X [Hor17] and the package [5XReleaseTools[105X.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (groupoids)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YMany-object structures[133X
    2.1 [33X[0;0YMagmas with objects; arrows[133X
      2.1-1 MagmaWithObjects
      2.1-2 IsDomainWithObjects
      2.1-3 Arrow
      2.1-4 IsSinglePieceDomain
    2.2 [33X[0;0YSemigroups with objects[133X
      2.2-1 SemigroupWithObjects
    2.3 [33X[0;0YMonoids with objects[133X
      2.3-1 MonoidWithObjects
    2.4 [33X[0;0YGenerators of magmas with objects[133X
      2.4-1 GeneratorsOfMagmaWithObjects
    2.5 [33X[0;0YStructures with more than one piece[133X
      2.5-1 UnionOfPieces
  3 [33X[0;0YMappings of many-object structures[133X
    3.1 [33X[0;0YHomomorphisms of magmas with objects[133X
      3.1-1 MagmaWithObjectsHomomorphism
    3.2 [33X[0;0YHomomorphisms of semigroups and monoids with objects[133X
    3.3 [33X[0;0YHomomorphisms to more than one piece[133X
      3.3-1 HomomorphismByUnion
      3.3-2 IsInjectiveOnObjects
    3.4 [33X[0;0YMappings defined by a function[133X
      3.4-1 MappingWithObjectsByFunction
  4 [33X[0;0YGroupoids[133X
    4.1 [33X[0;0YGroupoids: their properties and attributes[133X
      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 [33X[0;0YGroupoid elements; stars; costars; homsets[133X
      4.2-1 GroupoidElement
      4.2-2 IdentityArrow
      4.2-3 Order
      4.2-4 ObjectStar
    4.3 [33X[0;0YSubgroupoids[133X
      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 [33X[0;0YLeft, right and double cosets[133X
      4.4-1 RightCoset
    4.5 [33X[0;0YConjugation[133X
      4.5-1 \^
      4.5-2 ConjugateGroupoid
    4.6 [33X[0;0YGroupoids formed using isomorphisms[133X
      4.6-1 GroupoidByIsomorphisms
    4.7 [33X[0;0YGroupoids whose objects form a monoid[133X
      4.7-1 SinglePieceGroupoidWithRays
      4.7-2 RightActionGroupoid
  5 [33X[0;0YHomomorphisms of Groupoids[133X
    5.1 [33X[0;0YHomomorphisms from a connected groupoid[133X
      5.1-1 GroupoidHomomorphismFromSinglePiece
    5.2 [33X[0;0YProperties and attributes of groupoid homomorphisms[133X
      5.2-1 [33X[0;0YProperties of a groupoid homomorphism[133X
      5.2-2 [33X[0;0YAttributes of a groupoid homomorphism[133X
      5.2-3 RootGroupHomomorphism
      5.2-4 ImagesOfObjects
      5.2-5 ImageElementsOfRays
      5.2-6 ObjectGroupHomomorphism
    5.3 [33X[0;0YSpecial types of groupoid homomorphism[133X
      5.3-1 InclusionMappingGroupoids
      5.3-2 RestrictedMappingGroupoids
      5.3-3 IsomorphismNewObjects
      5.3-4 IsomorphismStandardGroupoid
      5.3-5 IsomorphismPermGroupoid
    5.4 [33X[0;0YHomomorphisms to a connected groupoid[133X
      5.4-1 HomomorphismToSinglePiece
      5.4-2 GroupoidHomomorphismFromHomogeneousDiscrete
    5.5 [33X[0;0YHomomorphisms to more than one piece[133X
      5.5-1 HomomorphismByUnion
      5.5-2 IsomorphismGroupoids
    5.6 [33X[0;0YAutomorphisms of groupoids[133X
      5.6-1 GroupoidAutomorphismByObjectPerm
      5.6-2 [33X[0;0YAutomorphisms of a groupoid with rays[133X
      5.6-3 AutomorphismGroupOfGroupoid
      5.6-4 [33X[0;0YInner automorphisms[133X
      5.6-5 GroupoidAutomorphismByGroupAutos
      5.6-6 AutomorphismGroupoidOfGroupoid
    5.7 [33X[0;0YMatrix representations of groupoids[133X
  6 [33X[0;0YGraphs of Groups and Groupoids[133X
    6.1 [33X[0;0YDigraphs[133X
      6.1-1 FpWeightedDigraph
    6.2 [33X[0;0YGraphs of Groups[133X
      6.2-1 GraphOfGroups
      6.2-2 IsGraphOfFpGroups
      6.2-3 RightTransversalsOfGraphOfGroups
    6.3 [33X[0;0YWords in a Graph of Groups and their normal forms[133X
      6.3-1 GraphOfGroupsWord
      6.3-2 ReducedGraphOfGroupsWord
    6.4 [33X[0;0YFree products with amalgamation and HNN extensions[133X
      6.4-1 FreeProductWithAmalgamation
      6.4-2 ReducedImageElm
      6.4-3 HnnExtension
    6.5 [33X[0;0YGraphsOfGroupoids and their Words[133X
      6.5-1 GraphOfGroupoids
      6.5-2 GraphOfGroupoidsWord
  7 [33X[0;0YTechnical Notes[133X
    7.1 [33X[0;0YMany object structures[133X
    7.2 [33X[0;0YMany object homomorphisms[133X
  8 [33X[0;0YDevelopment History[133X
    8.1 [33X[0;0YVersions of the Package[133X
    8.2 [33X[0;0YWhat needs to be done next?[133X
  
  
  [32X
