Circle
  
  
                         Adjoint groups of finite rings
  
  
                                 Version 1.6.1
  
  
                               25 September 2017
  
  
                              Alexander Konovalov
  
                               Panagiotis Soules
  
  
  
  Alexander Konovalov
      Email:    mailto:alexk@mcs.st-andrews.ac.uk
      Homepage: http://www.cs.st-andrews.ac.uk/~alexk/
      Address:  School of Computer Science
                University of St Andrews
                Jack Cole Building, North Haugh,
                St Andrews, Fife, KY16 9SX, Scotland
  
  
  Panagiotis Soules
      Email:    mailto:psoules@math.uoa.gr
      Address:  Department of Mathematics
                National and Capodistrian University of Athens
                Panepistimioupolis, GR-15784, Athens, Greece
  
  
  
  -------------------------------------------------------
  Abstract
  The  GAP4  package  Circle extends the GAP functionality for computations in
  adjoint  groups of associative rings. It provides functionality to construct
  circle  objects  that will respect the circle multiplication r ⋅ s = r + s +
  rs,  and  to  compute adjoint semigroups and adjoint groups of finite rings.
  Also  it  may  serve  as  an  example  of  extending the GAP system with new
  multiplicative objects.
  
  
  -------------------------------------------------------
  Copyright
  © 2006-2017 by Alexander Konovalov and Panagiotis Soules
  
  Circle  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
  http://www.gnu.org/licenses/gpl.html.
  
  If  you  obtained Circle, we would be grateful for a short notification sent
  to one of the authors.
  
  If  you  publish  a  result  which  was partially obtained with the usage of
  Circle, please cite it in the following form:
  
  A.  Konovalov, P. Soules. Circle --- Adjoint groups of finite rings, Version
  1.6.1; 2017 (http://www.cs.st-andrews.ac.uk/~alexk/circle/).
  
  
  -------------------------------------------------------
  Acknowledgements
  We  acknowledge  very  much  Alexander  Hulpke  and James Mitchell for their
  helpful  comments  and  advices, and the referee for testing the package and
  useful suggestions.
  
  
  -------------------------------------------------------
  
  
  Contents (Circle)
  
  1 Introduction
    1.1 General aims
    1.2 Installation and system requirements
  2 Implementing circle objects
    2.1 First attempts
    2.2 Defining circle objects
    2.3 Installing operations for circle objects
  3 Circle functions
    3.1 Circle objects
      3.1-1 CircleObject
      3.1-2 UnderlyingRingElement
      3.1-3 IsCircleObject
      3.1-4 IsPositionalObjectOneSlotRep
      3.1-5 CircleFamily
    3.2 Operations with circle objects
      3.2-1 One
      3.2-2 InverseOp
      3.2-3 IsUnit
      3.2-4 IsCircleUnit
    3.3 Construction of the adjoint semigroup and adjoint group
      3.3-1 AdjointSemigroup
      3.3-2 AdjointGroup
    3.4 Service functions
      3.4-1 InfoCircle
  4 A sample computation with Circle