{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

View Full Document Right Arrow Icon
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).
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}