The Boolean lattice of subsets of a finite set and its rank levels

For a finite set $A$, the Boolean lattice $B(A)$ is the finite power set $\mathcal P(A)$ ($\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$) ordered by inclusion. Its rank function is

$\rho(S):=|S|,$

and its rank-$k$ level is

$B(A)_k=[A]^k=\{\,S\subseteq A:|S|=k\,\}$

(The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$). Indeed, $T$ covers $S$ exactly when $T=S\cup\{a\}$ for one $a\in A\setminus S$, so a cover increases cardinality by one. The unique minimal element is $\varnothing$, of rank $0$, and hence $B(A)$ is graded (Graded poset, rank function, and rank levels).

If $|A|=n$, then the rank-$k$ level has cardinality $\binom nk$. The meet and join in this inclusion order are intersection and union, respectively.