MATH 681
Problem Set #6
This problem set is due at the beginning of class on
December 3
.
1.
(25 points)
Given ﬁnite posets (
S,
±
S
) and (
T,
±
T
), let us consider the set
S
×
T
subjected to the relation
±
which is such that (
a,b
)
±
(
c,d
) if and only if
a
±
S
c
and
b
±
S
d
.
(a)
(5 points)
Show that
±
is a partial ordering on (
S
×
T
).
(b)
(5 points)
Show that (
a,b
) is a maximal (or minimal) element of
S
×
T
if and
only if
a
and
b
are maximal (or minimal) elements of
S
and
T
respectively.
(c)
(5 points)
Show that if
C
is a chain in
S
×
T
, then the set
C
S
of ﬁrst coordinates
of elements of
C
is a chain in
S
. (similarly, it is true that the set
C
T
of second
coordinates of elements of
C
is a chain in
T
).
(d)
(5 points)
Demonstrate that if
A
is an antichain in
S
×
T
, then the sets
A
S
and
A
T
deﬁned as in the previous question need not be antichains.
(e)
(5 points + 5 points bonus)
If
S
has height
h
S
and width
w
S
, and
T
has height
h
T
and width
w
T
, what upper and lower bounds can you put on the height and
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.
 Fall '09
 WILDSTROM
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