The
*AbelianSystems* package is a
homalg
package (*GAP4*) which is dedicated to algebraic
systems theory for general classes of linear systems (defining an
abelian category) based on module
theory and homological algebra techniques.

The *OreModules* and *OreMorphisms* packages are *Maple*
prototypes which demonstrate the feasibility of a more professional
package dedicated to mathematical systems theory based on constructive approaches of module theory and homological algebra.

Using our experience, the goal of this project is to develop a *GAP4*
version of these packages built upon the powerful
homalg
package
developed by
M. Barakat (University of
Kaiserslautern) and his collaborators, and which is dedicated to homological
algebra oriented computations.

This task is simplified by the fact that the *homalg* package
already contains the implementation of the module theory and the homological
algebra techniques available in *OreModules* and *OreMorphisms*.
The *homalg* package also contains useful homological algebra
techniques which are not available in these packages (e.g.,
certain spectral sequences). Moreover, the *OreModules* package
can be directly called by the *homalg* package.

Using the friendly design of the *homalg* package, where the
different layers such as the computational engine (e.g., *Maple*,
*Singular*, *Macaulay2*, *Sage*, *MAGMA*), the module-theoretic results and the homological ones are separated, an efficient package dedicated to mathematical systems theory can now be developed in *GAP4*.

The main benefits of the future *AbelianSystems* package will be:

- Using the
*homalg*philosophy, inherited from*GAP4*, one could use its facilities to learn many pieces of information on a linear system, which were not directly asked by the user, during a particular computation. These properties will be stored and used to infer more properties on the linear systems through rules (theorems) as it is done in mathematics. - Using the independency of the
*homalg*package with respect to the computational engine (which, even between two computations, can be turned to*OreModules*,*JanetOre*,*Singular*,*Macaulay2*,*MAGMA*or*Sage*) and its interface with*Maple*, the efficiency of the new package will be much higher than the ones of the*OreModules*and*OreMorphisms*packages since the fastest engine can be used to handle a particular computation. Moreover, it will also give us the opportunity to use the last improvements and facilities of each computer algebra system.

The *AbelianSystems* package already contains the implementation of the purity
filtration computation of a
finitely generated module (which can also be computed by the
*homalg* package but by means of time-consuming spectral
sequences), and of the equidimensional block-triangular
representation of the corresponding linear system (as done
in the *PurityFiltration* package).

The *AbelianSystems* package is developed by
M. Barakat and A. Quadrat.