  
  
                                     [1X[5XCircle[105X[101X
  
  
                         [1XAdjoint groups of finite rings[101X
  
  
                                 Version 1.6.5
  
  
                                 27 April 2022
  
  
                               Olexandr Konovalov
  
                               Panagiotis Soules
  
  
  
  Olexandr Konovalov
      Email:    [7Xmailto:obk1@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://alex-konovalov.github.io/[107X
      Address:  [33X[0;14YSchool of Computer Science[133X
                [33X[0;14YUniversity of St Andrews[133X
                [33X[0;14YJack Cole Building, North Haugh,[133X
                [33X[0;14YSt Andrews, Fife, KY16 9SX, Scotland[133X
  
  
  Panagiotis Soules
      Email:    [7Xmailto:psoules@math.uoa.gr[107X
      Address:  [33X[0;14YDepartment of Mathematics[133X
                [33X[0;14YNational and Capodistrian University of Athens[133X
                [33X[0;14YPanepistimioupolis, GR-15784, Athens, Greece[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XGAP[105X4  package  [5XCircle[105X extends the [5XGAP[105X functionality for computations in
  adjoint  groups of associative rings. It provides functionality to construct
  circle  objects  that will respect the circle multiplication [22Xr ⋅ s = r + s +
  rs[122X,  and  to  compute adjoint semigroups and adjoint groups of finite rings.
  Also  it  may  serve  as  an  example  of  extending the [5XGAP[105X system with new
  multiplicative objects.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2006-2022 by Olexandr Konovalov and Panagiotis Soules[133X
  
  [33X[0;0Y[5XCircle[105X  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.      For      details,      see      the     FSF's     own     site
  [7Xhttp://www.gnu.org/licenses/gpl.html[107X.[133X
  
  [33X[0;0YIf  you  obtained [5XCircle[105X, we would be grateful for a short notification sent
  to one of the authors.[133X
  
  [33X[0;0YIf  you  publish  a  result  which  was partially obtained with the usage of
  [5XCircle[105X, please cite it in the following form:[133X
  
  [33X[0;0YO.  Konovalov, P. Soules. [13XCircle --- Adjoint groups of finite rings, Version
  1.6.5;[113X 2022 ([7Xhttps://gap-packages.github.io/circle/[107X).[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  acknowledge  very  much  Alexander  Hulpke  and James Mitchell for their
  helpful  comments  and  advices, and the referee for testing the package and
  useful suggestions.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Circle)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YGeneral aims[133X
    1.2 [33X[0;0YInstallation and system requirements[133X
  2 [33X[0;0YImplementing circle objects[133X
    2.1 [33X[0;0YFirst attempts[133X
    2.2 [33X[0;0YDefining circle objects[133X
    2.3 [33X[0;0YInstalling operations for circle objects[133X
  3 [33X[0;0Y[5XCircle[105X functions[133X
    3.1 [33X[0;0YCircle objects[133X
      3.1-1 CircleObject
      3.1-2 UnderlyingRingElement
      3.1-3 IsCircleObject
      3.1-4 IsPositionalObjectOneSlotRep
      3.1-5 CircleFamily
    3.2 [33X[0;0YOperations with circle objects[133X
      3.2-1 One
      3.2-2 InverseOp
      3.2-3 IsUnit
      3.2-4 IsCircleUnit
    3.3 [33X[0;0YConstruction of the adjoint semigroup and adjoint group[133X
      3.3-1 AdjointSemigroup
      3.3-2 AdjointGroup
    3.4 [33X[0;0YService functions[133X
      3.4-1 InfoCircle
  4 [33X[0;0YA sample computation with [5XCircle[105X[133X
  
  
  [32X
