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:

1. 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.

2. 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.

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