PROFESSOR: So we've seen that partial orders are
a set of axioms that capture the positive path relation
or the arbitrary path relation in directed acyclic graphs,
or DAGs.
But there's still another way to understand these axioms
that gives a kind of representation
theorem for the kind of mathematical objects
that are partial orders and that every partial order looks like.
So let's look at that example.
I'm interested in the proper subset relation.
A is a proper subset of B, which, you remember,
means that B has everything in it that A has and something
extra.
So in particular, since B has something extra,
B is not a subset of A, certainly not
a proper subset of A.
So let's look at an example of that.
Here are seven sets, and the arrows
indicate the proper subset relation.
Or more precisely, the positive path relation in this graph
represents the proper subset relation
where edges are understood to be pointing upwards.
So I've left out the arrowheads.
This is also known as a Hasse diagram,
where the height is an indication
of which way the arrows go.
So if arrows are pointing up, this
is telling me that, for example, this set of two elements,
1 and 5, because there's a path up to the top set,
the top set has everything that this lower set has.
Namely the top set has 1 and 5, and it's got extra stuff.
The set consisting of just 1 is a proper subset of 1 and 5
because the set has 1 in it, but it has an extra thing, 5.
And also, there's a path from 1 up to 1,
2, 5, 10 because 1, 2, 5, 10 has a 1 in it and extra stuff.
So that's what the picture is illustrating,
the proper subset relation on this particular collection
of seven sets.
Now, let's look at a very similar example
of the proper divides relation on some number.
So proper divides means a properly
divides b if a divides b and it's not equal to b.
And I'm interested in the proper divides relation
on this set of seven numbers, 1, 2, 3, 5, 10, 15, and 20.
And now there's a path from 5 to 30 because 5 is a divisor of 30
and it's not equal to 30.
It's a proper divisor of 30.
And of course, the point of this picture
is to show that the proper divides relation on these seven
numbers has exactly the same shape as the proper subset
relation on those seven sets.
So there's the seven sets and their proper subset relation
shown by the picture followed by the proper divides relation
on this set of seven numbers.
And the precise notion or sense in which
these things have the same shape--
obviously they can be drawn and one superimposed on the other.
But abstractly what we care about with partial orders
and digraphs in general is when things
are isomorphic-- is the technical name
for the same shape-- and isomorphic means
that all we care about are the connections
between corresponding vertices.
Two graphs where the vertices correspond in a way
that where there's a connection between two vertices
there's also a connection between the corresponding
vertices are isomorphic.
And the precise definition of isomorphic
is that they're isomorphic when there's
an edge-preserving matching between their vertices.
Matching means bijection.
And the formal definition is G1 is isomorphic to G2 if and only
if there's a bijection from V1, the vertices of G1,
to V2, the vertices of G2, with the property that if there's
an edge between two vertices u and v in the first graph, then
there's an edge between the corresponding two vertices
f of u and f of v in the second graph.
And that's an if and only if relation.
There's an edge between f of u and f of v if and only
if there's an edge between u and v in the original graph.
And that's the official definition
of isomorphism for digraphs.
And the theorem that we illustrated
with that example of proper divides and proper subset
is that, in fact, every strict partial order
is isomorphic to some collection of subsets partially ordered by
less than.
So this is a kind of a representation theorem.
If you want to know what kinds of things are partial orders,
the answer is that a strict partial order
is something that looks like a bunch of sets
under containment.
It's isomorphic to a bunch of sets under containment.
And the proof, actually, of this is quite straightforward.
What I'm going to do to find an isomorphism is you give me
your arbitrary strict partial order R,
and I'm going to map an element a in the domain of R
to the set of all of the elements that are,
quote, "below it," that is, all of the elements that
are related to R.
So a is going to map to the set of b's such
that bRa or b is equal to a.
And that is added for a technical condition.
Remember, R restrict, so a is not related to a under R.
But I want it to be in the set that a maps
to, so I'm throwing that in.
Another way to say this is that the mapping f of a
is equal to R inverse of a union a.
And let's just illustrate that by the example of,
how do you turn the divides relation into the subset
relation?
Well, the smallest element in the proper divides example
was the number 1, and I'm going to map it
to the set consisting of 1, which
is all of the elements that properly divide 1 along with 1.
And then I'm going to map the number
3 to all of the elements that properly divide 3 along with 3,
and that is 1 and 3.
5 maps to 1 and 5.
2 maps to 1 and 2.
And at the next level, I'm going to map 15
to all of the numbers that properly
divide 15 along with 15.
So 1, 3, 5, and 15 are what the number 15 maps to.
That's a set.
Likewise, 10 maps to 1, 2, 5, 15, and 30 maps
to all of the numbers that are below it, including itself.
And this is the general illustration of the way
that you take an arbitrary strict partial order and map
elements into sets, which are basically their inverse images
under the relation.
And the sets have exactly the same structure
under proper containment as the relation.
So this is, again, a representation theorem
that tells us that if we want to understand partial orders,
they are doing nothing more than talking about relations
with the same structure as the proper subset relation
on some collection of sets.