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# Circle

## Adjoint groups of finite radical algebras

Version 1.2

April 2007

Alexander Konovalov
e-mail: konovalov@member.ams.org
WWW: 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
e-mail: 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.

### Copyright

(C) 2006-2007 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 associative rings, Version 1.2; 2007 (http://www.cs.st-andrews.ac.uk/~alexk/circle.htm).

### Acknowledgements

We acknowledge very much Alexander Hulpke for helpful comments and advise.

### Contents

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
3.4-2 CIRCLEBuildManual
3.4-3 CIRCLEBuildManualHTML
4. A sample computation with Circle

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