The Circle Package: Adjoint groups of finite rings

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 GAP package ‘Circle’ is to extend 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.

Circle does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is redistributed with GAP, but is not loaded by default. Therefore, to use Circle, first you need to load it using the following command:

gap> LoadPackage("circle");

Olexandr Konovalov and Panagiotis Soules