GAP package SONATA

Authors: Erhard Aichinger, Franz Binder, Jürgen Ecker, Peter Mayr, Christof Nöbauer
Needs: GAP, version 4.4 or more recent
Optional: GAP package XGAP for graphic lattices
Operating systems: any
Current version: 2.4, released 20.11.2008
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.

  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.

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

Versions prior to 2.1: Download and unpack the archive given below in the root directory of your GAP installation.

Versions 2.1 and above: If you have installed GAP, version 4.4 and above, and if you have downloaded and unpacked the archive of all packages from the GAP website, 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.

Package Archive for Download

The SONATA package is availabe in the following formats:

An older version 2.0a for GAP 4.3 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; 2003. (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",
  year         = 2003,
  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.