maths.free › Combinatorics & Graph Theory › Partially Ordered Sets › The Subset Lattice

The Subset Lattice

When X is a finite set, the family of all subsets of X, partially ordered by inclusion, forms a subset latticeA lattice is a special type of poset.

The Subset Lattice

When \(X\) is a finite set, the family of all subsets of \(X\), partially ordered by inclusion, forms a subset latticeA lattice is a special type of poset. You do not have to concern yourself with the definition and can safely replace lattice with poset as you read this chapter.. We illustrate this in where we show the lattice of all subsets of \(\{1,2,3,4\}\). In this figure, note that we are representing sets by bit strings, and we have further abbreviated the notation by writing strings without commas and parentheses.

For a positive integer \(t\), we let \(\bftwo^t\) denote the subset lattice consisting of all subsets of \(\{1,2,\dots,t\}\) ordered by inclusion. Some elementary properties of this poset are:

The height is \(t+1\) and all maximal chains have exactly \(t+1\) points.
The size of the poset \(\bftwo^t\) is \(2^t\) and the elements are partitioned into ranks (antichains) \(A_0, A_1,\dots, A_t\) with \(|A_i|=\binom{t}{i}\) for each \(i=0,1,\dots,t\).
The maximum size of a rank in the subset lattice occurs in the middle, if \(s=\lfloor t/2\rfloor\), then the largest binomial coefficient in the sequence \(\binom{t}{0}, \binom{t}{1},\binom{t}{2},\dots,\binom{t}{t}\) is \(\binom{t}{s}\). Note that when \(t\) is odd, there are two ranks of maximum size, but when \(t\) is even, there is only one.

Sperner's Theorem

For the width of the subset lattice, we have the following classic result of Sperner.

Theorem: Sperner's Theorem

For each \(t\ge1\), the width of the subset lattice \(\bftwo^t\) is the maximum size of a rank, , \[\begin{aligned}\end{aligned}\].

Proof

The width of the poset \(\bftwo^t\) is at least \(C(t,\lfloor\frac{t}{2}\rfloor)\) since the set of all \(\lfloor\frac{t}{2}\rfloor\)-element subsets of \(\{1,2,\dots,t\}\) is an antichain. We now show that the width of \(\bftwo^t\) is at most \(C(t,\lfloor\frac{t}{2}\rfloor)\).

Let \(w\) be the width of \(\bftwo^t\) and let \(\{S_1,S_2,\dots, S_w\}\) be an antichain of size \(w\) in this poset, , each \(S_i\) is a subset of \(\{1,2,\dots,t\}\) and if \(1\le i\lt j\le w\), then \(S_i\nsubseteq S_j\) and \(S_j\nsubseteq S_i\).

For each \(i\), consider the set \(\cgS_i\) of all maximal chains which pass through \(S_i\). It is easy to see that if \(|S_i|=k_i\), then \(|\cgS_i|=k_i!(t-k_i)!\). This follows from the observation that to form such a maximum chain beginning with \(S_i\) as an intermediate point, you delete the elements of \(S_i\) one at a time to form the sets of the lower part of the chain. Also, to form the upper part of the chain, you add the elements not in \(S_i\) one at a time.

Note further that if \(1\le i \lt j\le w\), then \(\cgS_i\cap \cgS_j =\emptyset\), for if there was a maximum chain belonging to both \(\cgS_i\) and \(\cgS_j\), then it would imply that one of \(S_i\) and \(S_j\) is a subset of the other.

Altogether, there are exactly \(t!\) maximum chains in \(\bftwo^t\). This implies that \[\begin{aligned}\end{aligned}\]. This implies that \[\begin{aligned}\end{aligned}\] It follows that \[\begin{aligned}\end{aligned}\]. Thus \[\begin{aligned}\end{aligned}\].

À vous Aucune calculatrice ne tranche celle-ci, mais ses éléments sont calculables. Essayez-en un ci-dessous, ou tapez le vôtre.

Conservez votre propre travail

Un compte gratuit ajoute des notes sur chaque leçon, un enregistrement de ce que vous avez terminé, vos problèmes résolus en un seul endroit, et un tuteur que vous pouvez demander sur cette page. Les maths eux-mêmes sont ouverts à tout le monde, signé ou non.

Inscription Connexion

Symboles utilisés ici

Appuyez sur n'importe quel symbole pour la définition complète, une image et ce que signifie chaque lettre.

Questions posées par les gens

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Certaines parties de cette page sont adaptées de Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensés et réexpliqués ici ; les erreurs sont les nôtres.

Plus en Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Recurrences and generating functions Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton Colouring and planar graphs