submodule

• Examples
• User documentation
  • Operations
• Maintainer documentation for the classes module, and ModuleElem
• Bugs, Shortcomings and other ideas

Examples

• ex-module2.C

User documentation

Here are some pseudo-constructors for modules which are generated by a (finite) vector of ModuleElem (in a FreeModule). There is no class for submodules, they are objects of type FGModule.

There are several ways to create a submodule:

• submodule(M, gens) -- creates an FGModule representing the submodule of the FGModule M generated by the elements in gens; note that M must be specified even though it is usually implicit in the values contained in gens (unless gens is empty).
• SubmoduleCols(M, A) -- the submodule generated by the columns of the matrix A.
• SubmoduleRows(M, A) -- the submodule generated by the rows of the matrix A.

Operations

The permitted operations on submodules are:

    FGModule SyzOfGens(const FreeModule& F, const ideal& I);
    FGModule SyzOfGens(const FreeModule& F, const FGModule& N);
    FGModule SyzOfGens(const ideal& I);
    FGModule SyzOfGens(const FGModule& N);
    bool IsElem(const ModuleElem& v, const module& M);
    bool IsContained(const module& M, const module& M);

Maintainer documentation for the classes module, and ModuleElem

I shall suppose that the maintainer documentation for modules and FGModules has already been read and digested. It could also be helpful to have read ring.txt since the "design philosophy" here imitates that used for rings.

SubmoduleImpl is a concrete class derived from FGModuleBase, i.e. objects of this class represent submodules of explicitly finitely generated modules. The data members comprise the two obvious values:

    FreeModule myM;                 // the ambient module in which the generators live
    vector<ModuleElem> myGensArray; // the generators as specified by the user

Additionally there are two other data members:

    bool myTidyGensIsValid;             // true iff myTidyGensArray contains a correct value
    vector<ModuleElem> myTidyGensArray; // a "nice" set of generators

It is difficult to be precise about the value which myTidyGensArray should contain (when valid) since it depends upon the module. If the module is over a polynomial ring then it will be a Groebner basis. If the module is over Z then it will presumably be either a "Hermite Basis" or an "LLL Basis".

Bugs, Shortcomings and other ideas

Implementation and documentation are rather incomplete.

Why is myM a FreeModule and not an FGModule???

What is myTidyGensArray for a module over Z???