############################################################################# ## #W README.circle The CIRCLE package Alexander Konovalov ## Panagiotis Soules ## #H $Id: README.circle,v 1.6 2008/01/18 15:54:38 alexk Exp $ ## ############################################################################# The Circle Package ------------------ Let R be an associative ring, not necessarily with a unit element. The set of all elements of R forms a monoid with the neutral element 0 from R under the operation r*s = r + s + rs defined for all r,s from R. This operation is called the 'circle multiplication', and it is also known as the 'star multiplication'. The monoid of elements of R under the circle multiplication is called the adjoint semigroup of R. The group of all invertible elements of this monoid is called the adjoint group of R. These notions naturally lead to a number of questions about the connection between a ring and its adjoint group, for example, how the ring properties will determine properties of the adjoint group; which groups can appear as adjoint groups of rings; which rings can have adjoint groups with prescribed properties, etc. The main objective of the proposed GAP4 package 'Circle' is to extends the GAP functionality for computations in adjoint groups of associative rings to make it possible to use the GAP system for the investigation of such questions. Circle provides functionality to construct circle objects that will respect the circle multiplication r*s = r + s + rs, create multiplicative groups, generated by this objects, and compute groups of elements, invertible with respect to this operation, for finite radical algebras and finite associative rings without one. Installation and system requirements: Circle does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for GAP 4.4 and no compatibility with previous releases of GAP4 is guaranteed. To use the Circle online help it is necessary to install the GAP4 package GAPDoc by Frank Luebeck and Max Neunhoeffer, which is available from the GAP homepage or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/. Circle is distributed in standard formats (zoo, tar.gz, tar.bz2, -win.zip) and can be obtained the GAP homepage or from its homepage http://www.cs.st-andrews.ac.uk/~alexk/circle.htm. To unpack the archive circle-1.3.1. zoo you need the program unzoo, which can be obtained from the GAP homepage http://www.gap-system.org/ (see section `Distribution'). To install Circle, copy this archive into the 'pkg' subdirectory of your GAP4.4 installation. Then the subdirectory 'circle' containing the package will be created in the 'pkg' directory after the following command: unzoo -x circle-1.3.1.zoo Now you can start GAP and load the Circle package using the command: gap> LoadPackage("circle"); Alexander Konovalov and Panagiotis Soules January 2008