README.md
1[](https://travis-ci.org/gap-packages/circle)
2[](https://codecov.io/gh/gap-packages/circle)
3
4# The Circle Package: Adjoint groups of finite rings
5
6Let R be an associative ring, not necessarily with a unit element. The set
7of all elements of R forms a monoid with the neutral element 0 from R under
8the operation r*s = r + s + rs defined for all r,s from R. This operation is
9called the 'circle multiplication', and it is also known as the 'star
10multiplication'. The monoid of elements of R under the circle multiplication
11is called the adjoint semigroup of R. The group of all invertible elements of
12this monoid is called the adjoint group of R.
13
14These notions naturally lead to a number of questions about the connection
15between a ring and its adjoint group, for example, how the ring properties
16will determine properties of the adjoint group; which groups can appear as
17adjoint groups of rings; which rings can have adjoint groups with
18prescribed properties, etc.
19
20The main objective of the GAP package 'Circle' is to extend GAP functionality
21for computations in adjoint groups of associative rings to make it possible
22to use the GAP system for the investigation of such questions.
23
24Circle provides functionality to construct circle objects that will respect
25the circle multiplication r*s = r + s + rs, create multiplicative groups,
26generated by this objects, and compute groups of elements, invertible with
27respect to this operation, for finite radical algebras and finite associative
28rings without one.
29
30Circle does not use external binaries and, therefore, works without
31restrictions on the type of the operating system. It is redistributed with
32GAP, but is not loaded by default. Therefore, to use Circle, first you need
33to load it using the following command:
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35 gap> LoadPackage("circle");
36
37
38Alexander Konovalov and Panagiotis Soules
39
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