Type interface of AbstractAlgebra.jl

Apart from how we usually think of types in programming, we shall in this section discuss why we do not use the typical type interface.

Why types aren't enough

Naively, one might have expected that structures like rings in AbstractAlgebra.jl could be modeled as types and their elements as objects with the given type. But there are various reasons why this is not a good model.

Consider the ring $R = \mathbb{Z}/n\mathbb{Z}$ for a multiprecision integer $n$. If we were to model the ring $R$ as a type, then the type would somehow need to contain the modulus $n$. This is not possible in Julia, and in fact it is not desirable, since the compiler would then recompile all the associated functions every time a different modulus $n$ was used.

We could attach the modulus $n$ to the objects representing elements of the ring, rather than their type.

But now we cannot create new elements of the ring $\mathbb{Z}/n\mathbb{Z}$ given only their type, since the type no longer contains the modulus $n$.

Instead, the way we get around this in AbstractAlgebra.jl is to have special (singleton) objects that act like types, but are really just ordinary Julia objects. These objects, called parent objects, can contain extra information, such as the modulus $n$. In return, we associate this parent object with so called element objects.

In order to create new elements of $\mathbb{Z}/n\mathbb{Z}$ as above, we overload the call operator for the parent object.

In the following AbstractAlgebra.jl example, we create the parent object R corresponding to the ring $\mathbb{Z}/7\mathbb{Z}$. We then create a new element a of this ring by calling the parent object R.

R, = residue_ring(ZZ, 7)
a = R(3)

Here, R is the parent object, containing the modulus $7$. So this example creates the element $a = 3 \pmod{7}$.

Objects known as parents which contain additional information about groups, rings, fields and modules, etc., that can't be stored in types alone.

These details are technical and can be skipped or skimmed by new users of Julia/AbstractAlgebra.jl. Types are almost never dealt with directly when scripting AbstractAlgebra.jl to do mathematical computations.

In contrast, AbstractAlgebra.jl developers will want to know how we model mathematical objects and their rings, fields, groups, etc.

The abstract type hierarchy in AbstractAlgebra.jl

In AbstractAlgebra.jl, we use the abstract type hierarchy in order to give structure when programming the mathematical structures. For example, abstract types in Julia can belong to one another in a hierarchy.

For example, the Field abstract type belongs to the Ring abstract type. The full hierarchy can be seen in diagrams under the section on visualisation of the abstract types.

In practice this is practical since it means that any generic function designed to work with ring objects will also work with field objects.

In AbstractAlgebra.jl we also distinguish between the elements of a field, say, and the field itself.

For example, we have an object of type Generic.PolyRing to model a generic polynomial ring, and elements of that polynomial ring would have type Generic.PolyRingElem.

For this purpose, we also have a hierarchy of abstract types, such as FieldElem, that the types of element objects can belong to.

More complex example of parent objects

Here is some code which constructs a polynomial ring over the integers, a polynomial in that ring and then does some introspection to illustrate the various relations between the objects and types.

julia> using AbstractAlgebra

julia> R, x = ZZ["x"]
(Univariate polynomial ring in x over integers, x)

julia> f = x^2 + 3x + 1
x^2 + 3*x + 1

julia> R isa PolyRing

julia> f isa PolyRingElem

julia> parent(f) == R