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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 ⋅ 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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