Re: Lattice visualization tool

On Nov 14, 10:00 pm, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
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

You could have saved a lot of typing if you had just said "You're
dumb and your question is dumb." You could have saved even
more typing if you had simply not replied. All three possibilities
provide the OP with the same amount of assistance.

And here I've been wasting time thinking about lattice visualization
software for a few days. Oh if only I had known that since some
lattices are hard to render no one should ever consider rendering
any lattice.


Marshall
.

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