Re: Lattice visualization tool

On Wed, 14 Nov 2007, Feng wrote:

Hi, is there any easy-to-use software tools to visualize a lattice (or
a poset)? i.e., given a set of elements and order definition (in
certain format), the software is able to display a graph presenting
the relationship among the elements in an elegant way?

Yes, your mind's eye.
Computers can present only a few finite ordered sets.
Computers will do lousy presenting non-planar ordered sets.

Consider (N,|), positive intergers ordered by
x | y when x divides y.

It is a lattice with sup x,y = lcm x,y, inf x,y = gcd x,y

Now draw diagram for N up to 5 or 10. Use monospace font
and let the up direction denote greater, ie divides into.
For each two numbers, I've diagramed some of the sup's or lcm's.

8
/|
4 | 6
\|/|
2 3 5
\|/
1

Level 1 is 1.
Level 2 is primes
Level 3 is product of two primes
Level n is product of n - 1 primes

I've started a diagram for (N,|) with only three primes for level one.
Exercise.
Finish the diagram for (N,|) with only three primes at level one,
upto level 10. Take a look at the entangled view of this non-planar
diagram. Is such a diagram of any use?

Thus you'd left attempting to devise a method for visualizing
(N,|) that's simple and intuitively clear. In general, how to
diagram non-planar graphs onto a plane in an intuitive visual manner.

I suggest you consult with printed circuit and microchip programmers to
see what graphics they use to present or visualize their ever more a
jumbled tangles of connections.

See if their software can handle (N,|) for 10 or 100 primes at level one
upto level 100 or 1000. Of course (N,|) is finite. Another probem would
be displaying dense lattices, for example QxR with product order,
(q,x) <= (r,y) when q <= r, x <= y

.

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