Mathematical Nets - A Community Math Project
This is a community project, it's "Social Problem Solving". More minds make more progress!
If you feel you can contribute please email me your contribution. If I use it I will give you credit and a link to your website. Let's all advance this together!
Consider a set S and a binary operation C defined on S as follows..
C(a,b)= 0 or 1 where a and b are any two members of S.
If C(a,b)= 0 we say that a and b are not connected.
If C(a,b)= 1 we say that a is connected to b.
If C(b,a)= 1 we say that b is connected to a.
Notice that in general C is non-commutitive (non-abelian or non-symmetric).
We say that the operation C defines a net N over S and write N={S,C}. In general N={S,C} is a non-abelian net. A different choice of C will of course produce a different net.
Let's consider the special case of abelian nets where C(a,b)=C(b,a).
We define the "valence" of any member of N as the number of other members it connects to.
Conjecture
If every member of an abelian net has the same valence and that valence is even (valence=2*n where n=1,2,3..) then the net is an "n dimensional" system.
Next Steps
Question - How do we formally define dimension? Do we even need the concept? Is valence enough?
Question - Can the conjecture be proved?
Question - What happens when the valence is odd?
Content written and posted by Ken Abbott abbottsystems@gmail.com