{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

notes-091119

# notes-091119 - MATH 681 1 1.1 Notes Combinatorics and Graph...

This preview shows pages 1–2. Sign up to view the full content.

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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}