[Inquiry] Lattices As Charts

[Jon Awbrey]

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LAC.  Note 1

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Here is a first attempt to combine the ideas of lattice theory and
manifold theory for ontology applications.  (I was going to call
this "Atlases Of Lattices", but that acronym has been taken.)

o-------------------------------------------------------------o
| X                                                           |
|                                                             |
|              o-------------o   o-------------o              |
|             /               \ /               \             |
|            /                 o                 \            |
|           /                 / \                 \           |
|          /                 /   \                 \          |
|         /                 /     \                 \         |
|        o                 o   W   o                 o        |
|        |                 |   =   |                 |        |
|        |                 |U |^| V|                 |        |
|        |        U        |       |        V        |        |
|        |                 | x   y |                 |        |
|        |                 | o   o |                 |        |
|        o                 o       o                 o        |
|         \                 \     /                 /         |
|          \                 \   /                 /          |
|           \                 \ /                 /           |
|            \                 o                 /            |
|             \       |       / \       |       /             |
|              o------|------o   o------|------o              |
|                     |                 |                     |
|                     |                 |                     |
o---------------------|-----------------|---------------------o
                      |                 |
                    f |                 | g
                      |                 |
o---------------------|-----o     o-----|---------------------o
| L_1                 v     |     |     v                 L_2 |
|                           |     |                           |
|       o-----------o       |     |       o-----------o       |
|      / f(U)        \      |     |      /        g(V) \      |
|     /               o     |     |     o               \     |
|    /               / \    |     |    / \               \    |
|   /               /   \   |     |   /   \               \   |
|  o    ^  ^  ^    o fx  o  |     |  o gy  o    ^  ^  ^    o  |
|  |     \ | /     |^ ^ ^|  |     |  |^ ^ ^|     \ | /     |  |
|  |      \|/      | \|/ |  | !t! |  | \|/ |      \|/      |  |
|  |       o       |  o  ------------>  o  |       o       |  |
|  |      /|\      | /|\ |  |     |  | /|\ |      /|\      |  |
|  |     / | \     |v v v|  |     |  |v v v|     / | \     |  |
|  o    v  v  v    o  fy o  |     |  o  gx o    v  v  v    o  |
|   \               \   /   |     |   \   /               /   |
|    \               \ /    |     |    \ /               /    |
|     \               o     |     |     o               /     |
|      \             /      |     |      \             /      |
|       o-----------o       |     |       o-----------o       |
|                           |     |                           |
|                           |     |                           |
o---------------------------o     o---------------------------o

X is the manifold of what really exists.  It is called "X"
to serve as a constant reminder of how little we actually
know about anything of that description.  The chart maps
f and g are the maps that different parties of mappers
make of X, covering the domains U, V c X, respectively.

Accordingly, we have f : U -> L_1 and g : V -> L_2, where the domains of
f and g overlap in the intersection W = U |^| V that constitutes their
common domain, but f and g can form very different pictures of W.

In our application, let us assume that the chart codomains L_1 and L_2
are organized as lattices (partial orders, preorders, or what have you).
I have depicted a situation where f orders f(x) above f(y) in its lattice,
but g orders g(x) below g(y) in its lattice.

Notice that there is really no contradiction here, as these
orderings take place in two different "images" of X, but if
mappers unthinkingly "retro-project" their maps back onto X,
as they do when they insist that x >= y or x =< y, then there
will appear to be a conflict.

Jon Awbrey

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