When $X$ is a finite set, the family of all subsets of $X\text{,}$ partially ordered by inclusion, forms a subset lattice 1 . We illustrate this in Figure 6.26 where we show the lattice of all subsets of $\{1,2,3,4\}\text{.}$ 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 $t\text{,}$ 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:

1. The height is $t+1$ and all maximal chains have exactly $t+1$ points.

2. 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 $i=0,1,\dots,t\text{.}$

3. The maximum size of a rank in the subset lattice occurs in the middle, i.e. if $s=\lfloor t/2\rfloor\text{,}$ then the largest binomial coefficient in the sequence $\binom{t}{0}, \binom{t}{1},\binom{t}{2},\dots,\binom{t}{t}$ is $\binom{t}{s}\text{.}$ Note that when $t$ is odd, there are two ranks of maximum size, but when $t$ is even, there is only one.