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5.3
From Partial to Total Orders
Given a finite partial order, we can extend it to a total order. For example,
suppose we have a set of tasks
T
to perform. We wish to decide in what
order to perform them.
We are not totally free to choose, because some
tasks have to be finished before others can be started. We can express this
pre-requisite structure by a partial order
<
on
T
. We want to find a total
order
<
on
T
which respects
<
in the sense that if
t < u
then
t < u
.
As a more concrete example, consider the partial order
⊆
on
P
(
{
1
,
2
,
3
}
)
given in example 5.5. It is partial because, for example, the sets
{
1
}
and
{
3
}
are not contained in each other. A total order
⊆
T
which extends this partial
order is given by the sequence
∅
,
{
3
}
,
{
2
}
,
{
1
}
,
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
1
,
2
,
3
}
We are forced to have
{
1
}
⊆
T
{
1
,
2
}
since we wish to respect the partial
order, but we have chosen to have
{
3
}
⊆
T
{
1
}
. This process of going from
a partial to a total order is called
topological sorting
, and we can define a
simple algorithm for topological sorting based on minimal elements.
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- Spring '09
- Koskesh
- Math, Partially ordered set, Total order
-
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