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#W README.circle The CIRCLE package Alexander Konovalov
## Panagiotis Soules
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The Circle Package
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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. The current version is
designed for GAP 4.5 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 (tar.gz, tar.bz2, zip, -win.zip)
and can be obtained the GAP homepage or from its homepage
http://www.cs.st-andrews.ac.uk/~alexk/circle/. To install the package, unpack
its archive in the 'pkg' subdirectory of your GAP4.5 installation. Now you
can start GAP and load the Circle package using the command:
gap> LoadPackage("circle");
Alexander Konovalov and Panagiotis Soules May 2012