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PROFESSOR: Partial
orders are another way

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to talk about digraphs
and they offer to us

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an interesting
lesson in the idea

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of axiomatizing a
mathematical structure

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and mathematical ideas.

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So let's begin by discussing
some of the properties

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that we're going to use to
axiomatize partial orders

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and digraphs.

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So if we think about
walks in a digraph,

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the basic property of walks is
that if you have a walk from u

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to v and you have
a walk from v to w,

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then you put the
two walks together

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and you wind up with
a walk from u to w.

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Expressed in terms of the
positive walk relation

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in G, what this is saying is
that if u G plus v and v G

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plus w, then u G plus w.

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And that abstract
property-- which

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I'm highlighting with
the magenta box--

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when you apply it to
an arbitrary relation

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is called a
transitivity property.

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So a relation R on
a set-- that is,

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R is relating-- the
domain and co-domain of R

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are the same-- is that u R
v and v R w implies u R w.

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And a relation that
has that property

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is said to be transitive.

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And, of course,
what we've just seen

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is the positive path relation
of any graph G is transitive.

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Another way to say
transitivity is

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to read u R v as saying
there's an edge from u to v.

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And what this says is that if
there's an edge from u to v

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and an edge from v to w,
there's an edge from u to w.

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Or, in other words, if
there's a path of length 2,

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there's a path of length 1.

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And then by easy
induction it follows

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that if there's a
path of any length

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between-- of any positive
lengths between two vertices,

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then in fact there's
a path of length 1.

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That is, an edge between them.

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OK, so the basic theorem
that we have to begin with

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is what is
transitivity capturing

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as a property of a relation.

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And a relation R
is transitive if

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and only if, in fact, R is
equal to the positive walk

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relation for some digraph
G. The proof of this

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is basically trivial
because you could

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let the relation
R be the digraph

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that it's the positive
path relation of.

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If we look now at
directed acyclic graphs,

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then what we have is that
if there's a positive length

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path from a vertex
u to a vertex v,

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then since there's no cycles
in a directed acyclic graph,

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there can't be a path
back from v to u,

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and that property
is called asymmetry.

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So D plus, which is the
positive path relation, in a DAG

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has this asymmetry property.

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Namely, if u can get to v by a
positive length path and it's

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not possible for v to get back
to u by a positive length path.

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So, abstracted, u R
v implies not v R u.

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That's the asymmetry property
of an arbitrary relation R.

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And by definition of acyclic,
D plus is asymmetric in a graph

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without cycles.

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OK.

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A strict partial order
is simply a relation

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that has these two
properties of being

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transitive and asymmetric.

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And some examples of
strict partial orders

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are the proper
containment relation

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on sets, which we've
previously commented

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can be viewed as a DAG, but
now it satisfies transitivity.

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And the fact that if one set's
properly contained in another,

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the second one can't be
properly contained in the first

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because proper means you
have something extra.

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The indirect prerequisite
relation on MIT subjects

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would be another example
of a strict prerequisite.

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If I'm a prerequisite
of you, you

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can't be a prerequisite of me.

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And finally, the less than
relation on real numbers.

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These are all examples
of strict partial orders.

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And putting together
the previous reasoning,

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what we can say is
that a relation R

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is a strict partial
order if and only

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if R is the positive path
relation for some DAG, D.

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So the axioms that define
strict partial order,

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namely transitivity
and asymmetry,

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can be said to
abstractly capture

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the property of a relation
that it comes from a DAG.

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Another important
property of partial orders

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is the idea of being
path-total, or linear

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as some authors call it.

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And the simple
definition of path-total

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is that given any
two elements, one

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is going to be bigger than
the other with respect

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to the relation.

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Most familiar example of that
would be the less than relation

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of a less than or equal
to relational on the reals

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given any two distinct
real numbers-- x and y.

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Either x is less than
y or y is less than x.

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And we take that
property for granted.

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Now, the formal
definition then is simply

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that if x is not equal to y,
then either x R y or y R x,

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and relation R that has that
property is called path-total.

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Another way to say
it is that there

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are no incomparable
elements under R.

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And I've, again, highlighted
with a magenta box

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this property, which is
called path-totality.

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Another way to say
that a path-total is

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that the whole order
looks like a chain.

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If you give me a
bunch of elements,

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there's going to have to
be a biggest one and then

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a next biggest one and so
on, assuming you've given me

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any finite set of elements.

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So the basic example,
again, of path-total

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would be number
properties of bigger than.

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And a basic example of something
that would typically not

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be path-total would
be, let's say,

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subset containment where
you can perfectly well have

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two sets, neither of which
is contained in the other.

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So a weak partial order
is a small variation

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of a strict partial order.

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That is another
familiar concept where

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we take the strict
property, which

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guarantees that nothing's
related to itself,

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and we relax it.

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So a strict partial
order is just

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like a weak partial order
except that the condition

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that there's no positive
length path between an element

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in itself is relaxed.

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So, in fact, it's
not only relaxed,

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but it's completely denied.

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In a weak partial order, we
insist that every element

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is related to itself.

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An example of that would be the
less than or equal to relation.

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Sorry, the improper
containment relation.

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The ordinary subset
relation on sets

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where now a is a subset
with a bar under it.

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a is just a subset
of a, not necessarily

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a strict subset
or a proper subset

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means that, in fact,
a is a subset of a.

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And then less than or equal,
and you put the little bar

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under the less than sign to
indicate that equality is also

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a possibility, you get
a weak partial order

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on the real numbers.

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So the property that
distinguishes the weak

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from the strict is this
property of reflexivity.

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A relation R on a
set is reflexive

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if every element is
related to itself,

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if and only if a R a for all
little a in the domain capital

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A. And what we can
observe immediately

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is that the path of the walk
relation-- G star-- which

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includes walks of
length zero is reflexive

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because, by definition,
there is a length zero

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walk from any vertex to itself.

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So if you're going
to play with axioms,

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then you can
reformulate asymmetry--

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the idea of asymmetry
except for elements

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being related to themselves.

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It's called
antisymmetry, and it says

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that a relation R
is antisymmetric if

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and only if it's asymmetric
except for the a R a case.

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And more precisely,
the difference

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between asymmetry
and antisymmetry

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is that in asymmetry a
R a is never allowed,

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and in antisymmetry a
R a is a possibility.

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It's not disallowed.

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So an antisymmetric relation
on R stated abstractly

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is that u R v implies not
v R u for u not equal to v.

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So the first line is exactly
the statement of asymmetry,

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and then I add this
proviso that it only

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has to hold when the u
and the v are not equal.

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That's the formal way
of saying antisymmetry

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is the same as asymmetry
except for a R a.

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And the walk relation
in a digraph,

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which includes length zero
walks, is antisymmetric.

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So weak partial
orders-- just what

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you get when you put
these things together--

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weak partial order is
transitive, antisymmetric,

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and reflexive.

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So in a weak partial order,
we insist that every element

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be related to itself.

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So there's a-- just
a quick remark.

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Asymmetric implies
nothing's related to itself.

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Reflexive implies everything
is related to itself.

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And it's possible that
there be some graph

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in which some
elements are related

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to themselves and some not.

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That would be something
that was neither

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a strict nor a
weak partial order.

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It would just be transitive
and antisymmetric.

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Those don't come
up much and so we

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don't bother to give them
a name or talk about them.

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And, finally, the theorem that
summarizes up this whole story

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is that R is a weak
partial order if and only

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if R is equal to the walk
relation for some DAG,

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including length zero walks.