Description:

SONATA stands for "systems of nearrings and their applications". It
provides methods for the construction and the analysis of finite
nearrings. A left nearring is an algebra (N;+,*), where (N,+) is a (not
necessarily abelian) group, (N,*) is a semigroup, and x*(y+z) = x*y + x*z
holds for all x,y,z in N.

As a typical example of a nearring, we may consider the set of all
mappings from a group G into G, where the addition is the pointwise
addition of mappings in G, and the multiplication is composition of
functions. If functions are written on the right of their arguments,
then the left distributive law holds, while the right distributive law
is not satisfied for non-trivial G.

The SONATA package provides methods for the construction and analysis of
finite nearrings.
1. Methods for constructing all endomorphisms and all fixed-point-free
automorphisms of a given group.
2. Methods for constructing the following nearrings of functions on a
group G:
- the nearring of polynomial functions of G (in the sense of
Lausch-Nöbauer);
- the nearring of compatible functions of G;
- distributively generated nearrings such as I(G), A(G), E(G);
- centralizer nearrings.
3. A library of all small nearrings (up to order 15) and all small
nearrings with identity (up to order 31).
4. Functions to obtain solvable fixed-point-free automorphism groups on
abelian groups, nearfields, planar nearrings, as well as designs from
those.
5. Various functions to study the structure (size, ideals, N-groups, ...)
of nearrings, to determine properties of nearring elements, and to
decide whether two nearrings are isomorphic.
6. If the package XGAP is installed, the lattices of one- and two-sided
ideals of a nearring can be studied interactively using a graphical
representation.