# [GAP Forum] Refereed Package SONATA

Steve Linton sal at dcs.st-and.ac.uk
Fri Jun 18 11:15:12 BST 2004

```Dear GAP Forum,

The SONATA package for "systems of nearrings and their applications" was
accepted through the package refereeing process in 2003, but its release was
delayed for technical reasons until the release of GAP 4.4
a few months ago, and as responsible editor, I am afraid I qforgot to send the
customary message announcing its acceptance and release at that time.

So, I can now belatedly, but with great pleasure, announce that the SONATA
package, by Erhard Aichinger, Franz Binder, Juergen Ecker, Peter Mayr and
Christof Noebauer has been accepted as a refereed GAP package and is available
for download from the GAP Web, and FTP sites, or from the authors Web page at
http://www.algebra.uni-linz.ac.at/Sonata/. Note however, that anyone who has
SONATA.

The following brief description of the functionality of the package is taken
from the authors' Web page.

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.

I apologise again for the delay in this announcement,

Steve Linton

--
Steve Linton	School of Computer Science  &
Centre for Interdisciplinary Research in Computational Algebra
University of St Andrews 	 Tel   +44 (1334) 463269
http://www.dcs.st-and.ac.uk/~sal	 Fax   +44 (1334) 463278

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