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# Circle

## Adjoint groups of finite rings

Version 1.6.5

27 April 2022

Olexandr Konovalov
Email: obk1@st-andrews.ac.uk
Homepage: https://alex-konovalov.github.io/
Address:
School of Computer Science
University of St Andrews
Jack Cole Building, North Haugh,
St Andrews, Fife, KY16 9SX, Scotland

Panagiotis Soules
Email: 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 \( r \cdot s = r + s + rs \), 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

© 2006-2022 by Olexandr 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:

O. Konovalov, P. Soules. Circle --- Adjoint groups of finite rings, Version 1.6.5; 2022 (https://gap-packages.github.io/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

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
References
Index
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