[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 -------------------------------------------------------