  
  
                                     [1X[5XCircle[0m[1X[0m
  
  
                         [1XAdjoint groups of finite rings[0m
  
  
                                 Version 1.3.1
  
  
                                  January 2008
  
  
                              Alexander Konovalov
  
                               Panagiotis Soules
  
  
  
  Alexander Konovalov
      Email:    [7Xmailto:konovalov@member.ams.org[0m
      Homepage: [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/[0m
      Address:  School of Computer Science
                University of St Andrews
                Jack Cole Building, North Haugh,
                St Andrews, Fife, KY16 9SX, Scotland
  
  
  Panagiotis Soules
      Email:    [7Xmailto:psoules@math.uoa.gr[0m
      Address:  Department of Mathematics
                National and Capodistrian University of Athens
                Panepistimioupolis, GR-15784, Athens, Greece
  
  
  
  -------------------------------------------------------
  [1XAbstract[0m
  The  [5XGAP[0m4  package  [5XCircle[0m extends the [5XGAP[0m 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.
  
  
  -------------------------------------------------------
  [1XCopyright[0m
  © 2006-2008 by Alexander Konovalov and Panagiotis Soules
  
  [5XCircle[0m  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[0m.
  
  If  you  obtained [5XCircle[0m, 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
  [5XCircle[0m, please cite it in the following form:
  
  A.  Konovalov, P. Soules. [13XCircle --- Adjoint groups of finite rings, Version
  1.3.1;[0m 2008 ([7Xhttp://www.cs.st-andrews.ac.uk/~alexk/circle.htm[0m).
  
  
  -------------------------------------------------------
  [1XAcknowledgements[0m
  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.
  
  
  -------------------------------------------------------
  
  
  [1XContents (Circle)[0X
  
  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 [5XCircle[0m 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
      3.4-2 CIRCLEBuildManual
      3.4-3 CIRCLEBuildManualHTML
  4 A sample computation with [5XCircle[0m
  
  
  -------------------------------------------------------
