The spectrum of a finite pseudocomplemented lattice

Let L be a finite pseudocomplemented lattice. Every interval 0, a] in L is pseudocomplemented, so by Glivenko’s theorem, the set S(a) of all pseudocomplements in 0, a] forms a boolean lattice. Let B _i denote the finite boolean lattice with i atoms. We describe all sequences (s ₀, s ₁, . . . , s _n ) of integers for which there exists a finite pseudocomplemented lattice L with s _i = |{ a ∈ L | S(a) ? B _i }|, for all i, and there is no a ∈ L with S(a) ? B _n+1. This result settles a problem raised by the first author in 1971.