GAP package SONATA
Authors:
Erhard Aichinger,
Franz Binder,
Jürgen Ecker,
Peter Mayr,
Christof Nöbauer
Needs: GAP,
version 4.5 or more recent
Optional:
GAP package XGAP for graphic lattices
Operating systems: any
Current version: 2.8, released 29.07.2015
Status: accepted
Contact:
email
Homepage:
www.algebra.uni-linz.ac.at/Sonata/
Download:
.tar.gz format
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.
- Methods for constructing all endomorphisms and all
fixed-point-free automorphisms of a given group.
- 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.
- A library of all small
nearrings (up to order 15) and all small nearrings with identity (up
to order 31).
- Functions to obtain solvable fixed-point-free automorphism
groups on abelian groups, nearfields, planar nearrings, as well
as designs from those.
- 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.
- 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.
References
J.R. Clay, Nearrings: Geneses and applications, Oxford University Press
-- Oxford, New York, Tokyo, 1992.
C. Cotti Ferrero and G. Ferrero, Nearrings. some developments linked to
semigroups and groups, Kluwer Academic Publishers, Dordrecht, Boston,
London, 2002.
H. Lausch and W. Nöbauer, Algebra of polynomials, North-Holland,
Amsterdam, London; American Elsevier Publishing Company, New York, 1973.
J.D.P. Meldrum, Near-rings and their links with groups, Pitman
(Advanced Publishing Program), Boston, Mass., 1985.
G.F. Pilz, Near-rings, 2nd ed., North-Holland Publishing Company --
Amsterdam, New York, Oxford, 1983.
H. Wähling, Theorie der Fastkörper, Thales-Verlag, Essen, 1987.
Installation
If you have installed
GAP, version 4.5 and above,
then you have already installed SONATA.
You can also obtain SONATA as part of the optional
packages for SAGE from the
SAGE website.
Otherwise download and unpack the
archive given
below in the pkg
directory of your
GAP installation.
Online Documentation
The manual of the package is available in several formats.
If the package has been installed properly, the manuals
are also accessible through the GAP online
help. In addition, a tutorial for the package is available.
The SONATA package is availabe in the following
formats:
Older versions can also be downloaded:
Citation
If you use SONATA in the preparation of a paper, please cite
it in the following form.
E. Aichinger, F. Binder, J. Ecker, P. Mayr, and C. Nöbauer SONATA - system of near-rings and their applications, GAP package, Version 2.8; 2015.
(http://www.algebra.uni-linz.ac.at/Sonata/)
If you are using BibTeX, you can use the following BibTeX entry.
@manual{SONATA,
author = "Aichinger, E. and Binder, F. and Ecker, J. and Mayr, P. and N{\"o}bauer, C.",
title = "{SONATA} - system of near-rings and their applications, {GAP} package, Version 2.8",
year = 2015,
note = "\verb+(http://www.algebra.uni-linz.ac.at/Sonata/)+"}
Acknowledgements
SONATA was written at the Department of Algebra of
the University of Linz, Austria.
The development of SONATA was kindly supported by the
Austrian Science Fund (FWF) within the projects P11486-TEC
and P12911-INF.