The spectrum of a finite pseudocomplemented modular lattice

We propose the minimal representation of an element in a pseudocomplemented modular lattice satisfying the descending chain condition and prove that the minimal representation of an element is unique. Then we give necessary and sufficient conditions under which a sequence can be the spectrum of a modular lattice and a distributive lattice, respectively. These results settle two problems raised by G. Grätzer, D. S. Gunderson and R. W. Quackenbush in 2009.