SUMMARY
Let [n]={1,2,…,n}[n]=\{1,2,\ldots,n\} and Bn={A:A?[n]}\mathscr{B}_n=\{A: A\subseteq [n]\}. A family A?Bn\mathscr{A}\subseteq \mathscr{B}_n is a Sperner family if A?A\nsubseteq B and B\nsubseteq AB\nsubseteq A for distinct A,B\in\mathscr{A}A,B\in\mathscr{A}. Sperner's theorem states that the density of the largest Sperner family in \mathscr{B}_n\mathscr{B}_n is \binom{n}{\left\lceil{n/2}\right\rceil}/2^n\binom{n}{\left\lceil{n/2}\right\rceil}/2^n. The objective of this note is to show that the same holds if \mathscr{B}_n\mathscr{B}_n is replaced by compressed ideals over [n][n].