Problem 1
(#8.6.8)
.
Determine whether the relations represented by the following
zeroone matrices are partial orders:
a)
1 0 1
1 1 0
0 0 1
b)
1 0 0
0 1 0
1 0 1
c)
1 0 1 0
0 1 1 0
0 0 1 1
1 1 0 1
Solution.
By inspection, all three matrices are reﬂexive and antisymmetric, so the
only thing we have to check in each case is transitivity. Relation (a) is not transitive,
since the second element relates to the ﬁrst, and the ﬁrst to the third, but the second
element doesn’t relate to the third. Relation (b) is transitive, and represents the
partial order 3
±
1. Relation (c) also isn’t transitivethe ﬁrst element relates to the
third, and the third to the fourth, but the ﬁrst doesn’t relate to the fourth.
Problem 2
(#8.6.10)
.
Determine whether the relation with the shown directed
graph is a partial order.
Solution.
The graph shown is almost that of the total ordering
a
±
c
±
d
±
b
,
but it’s missing
c
±
b
. Since it isn’t transitive, it’s not the digraph of a partial
ordering.
Problem 3
(#8.6.12)
.
Let
(
S,R
)
be a poset. Show that
(
S,R

1
)
is also a poset,
where
R

1
is the inverse of
R
. The poset
(
S,R

1
)
is called the dual of
(
S,R

1
)
.
Proof.
We must show that
R

1
is a partial ordering, given that
R
is. Since (
a,a
)
∈
R
, (
a,a
)
∈
R

1
,so
R

1
is reﬂexive. Suppose (
a,b
)
∈
R

1
and (
b,a
)
∈
R

1
. Then
(
b,a
)
∈
R
and (
a,b
)
∈
R
, so
a
=
b
. Thus,
R

1
is also antisymmetric. Finally,
suppose (
a,b
)
∈
R

1
and (
b,c
)
∈
R

1
. Then (
b,a
)
∈
R
and (
c,b
)
∈
R
, so by the
transitivity of
R,
(
c,a
)
∈
R
. Thus (
a,c
)
∈
R

1
, and
R

1
is also transitive. Thus
R

1
is a partial ordering, and
S,R

1
is a poset, as claimed.
Problem 4
(#8.6.28)
.
Let
(
S,
±
)
be a poset. We say that an element
y
∈
S
covers
an elements
x
∈
S
if
x
≺
y
and there is no element
z
∈
S
such that
x
≺
z
≺
y
.
The set of pairs
(
x,y
)
such that
y
covers
x
is called the covering relation of
(
S,
±
)
.
What is the covering relation of the partial ordering
{
(
a,b
)

a
divides
b
}
on
{
1
,
2
,
3
,
4
,
6
,
12
}
?
Solution.
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 Spring '11
 MichealAlain
 Math, Graph Theory, Matrices, Vertex, Order theory, loop, Total order

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