[kagc] Module Structure
- From: Geoffrey Platta <platta_geoffrey@xxxxxxxxxxxxxxxxx>
- To: kagc@xxxxxxxxxxxxx
- Date: Fri, 6 Jun 2014 18:34:50 -0400
Before developing the API, first the understanding of what a module is
should be developed. A module should probably best be built off topics in
analysis and abstract algebra.
In algebras, the following concepts are important: sets, groups, rings, and
fields.
We've gone over sets in both discrete math classes, and if you are taking
linear algebra you will probably cover it in more depth than in discrete.
It is an intuitive enough concept, and very important in all higher maths.
An example of a set would be the set of all numbers between zero and one.
All numbers between zero and one can be thought of as an infinite sequence
of numbers where no members of the set have the same sequence. A version of
this example--actually, *[0,1]x[0,1]*, the Cartesian product of all numbers
between zero and one*--*was used in my differential geometry text, and I
did not expect this sort of understanding and so didn't come close to
understanding the example.
A group is a set of elements which, when an operation is applied, results
in an element within the group. For example, the real numbers and addition
form a group; you can never perform an addition which gives a non-real
number. Real numbers and multiplication also form a group; the reals and
division do not, because division by zero is undefined; additionally, real
numbers and exponentiation do not form a group, because the even roots of
negative numbers have complex results. More examples of groups are modulus
arithmetic, in which addition and multiplication yield results in [0,n) in
the set *x mod n *for any x.
A ring is a group in which addition and multiplication are defined. The
complex numbers form a ring, as do all subsets of complex numbers (such as
the reals *R*, rationals *Q*, integers *Z*, etc).
Anyway, the point is, I am thinking that there might need to be different
sorts of modules. Algebraic modules could define sets and/or their
operations, whether the set and the operation form a group or not. Calculus
modules could define rates of change in algebraic modules. Unfortunately
the difference between a calculus and an algebra seems to be poorly
defined, so defining a calculus module as describing rates of change within
an algebra is my own definition and probably not accurate.
Geometries and topologies depend heavily on abstract algebra, and
differential geometry and differential topology involve the application of
calculus to the sets in a geometric or topological "structure".
But yeah, what a module describes should be more or less set in stone
before the API is started.
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