MATH 681
Notes
Combinatorics and Graph Theory I
1
Chains and Antichains
1.1
Maximality and maximum, uh, ness?
To review, the definitions of a chain and antichain:
Definition 1.
A
chain
is a totally ordered subset of a poset
S
; an
antichain
is a subset of a poset
S
in which any two distinct elements are incomparable.
Now, we have two distinct concepts of a chain/antichain being “as large as possible”. One of those
concepts is “not easily extendable” – that is to say, a chain or antichain which can’t have elements
added to make it a larger chain/antichain:
Definition 2.
A
maximal
chain (antichain) is one that is not a proper subset of another chain
(antichain).
Alternatively, a chain
C
is
maximal
in a poset (
S,
) if no element of
S

C
is comparable to every
element of
C
; an antichain
A
is
maximal
if no element of
S

A
is incomparable to every element
of
A
.
Our other definition of largeness is that of simply being of the greatest size in a poset.
Definition 3.
A
maximum
or
longest chain
(
largest antichain
) is one which is of the greatest size
possible. The size of the longest chain is known as a poset’s
height
. The size of the largest antichain
is known as a poset’s
width
.
Note, trivially, that a maximum chain/antichain must be maximal: if
C
is maximum, than there
are no chains of size greater than

C

; if
C
were a proper subset of a chain
D
, then

D

>

C

would
contradict the above assertion.
As an example, let us consider the poset (
{
1
,
2
,
3
,
4
,
5
,
6
,
9
,
12
,
18
}
,

), recalling that
a

b
is
a
is a
divisor of
b
.
Then we may note that
{
1
,
2
,
4
,
12
}
is a maximum chain — there are no chains of length greater
than 4. The maximum chain is not unique:
{
1
,
3
,
6
,
18
}
is also a maximum chain, as is
{
1
,
2
,
6
,
12
}
and several others. An example of a maximal chain which is
not
maximum is
{
1
,
5
}
: this has only
two elements, and thus is not maximum, but no more elements of
S
can be added to it, since 5
is incomparable with everything except 1 and 5.
An example of a nonmaximal chain would be
{
1
,
3
,
18
}
, because it is a subset of a larger chain (in fact, two larger chains, since either 6 or 9 could
be included).
Looking at antichains, the largest we could find would have size 4:
{
4
,
6
,
9
,
5
}
would be an example,
and is in fact the unique maximum antichain (uniqueness is not guaranteed; it’s just how this
example works out). We could find several maximal antichains which are not maximum, such as
{
2
,
3
,
5
}
or
{
12
,
18
}
, or even
{
1
}
(which would be maximal because everything is comparable to 1!).
And of course, nonmaximal antichains can be found too, such as
{
12
,
5
}
, which is a subset of the
larger antichain
{
12
,
9
,
5
}
.
There are two useful simple observations to be made about maximality, one for chains, and one for
antichains.
Proposition 1.
A maximal chain in a finite nonempty poset must contain a maximal element of
S
(and a minimal element).
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 Fall '09
 WILDSTROM
 Combinatorics, Graph Theory, Order theory, Partially ordered set, Antichain, antichains

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