The spectrum of a finite pseudocomplemented lattice
Let \(L\) be a pseudocomplemented lattice, then every interval \([0,a]\) of \(L\) is also pseudocomplemented. So, by Glivenko's theorem, the set \(S(a)\) of all pseudocomplements in \([0,a]\) forms a Boolean lattice. Let \(L\) be a finite pseudocomplemented lattice and suppose that \(S(1)\) has exactly \(n\) atoms. Let \(B_i\) denote the finite Boolean algebra with \(i\) atoms, then the spectrum of \(L\) is the sequence \((s_0,x_1,\dots, s_n)\), where \(s_i=|\{a\in L\mid S(a)\cong B_i\}|\). Clearly, \(s_0+ s_1+\cdots+ s_n=|L|\) and \(s_0= 1\). The main result of the paper is the following theorem: A sequence \((1,s_1,\dots, s_n)\) of positive integers is the spectrum of a finite pseudocomplemented lattice if and only if the inequality \({n\choose i}\leq s_i\) holds for all \(1\leq i\leq n\). This result solves a problem raised in \textit{G. Grätzer}'s book [Lattice theory. First concepts and distributive lattices. San Francisco: Freeman (1971; Zbl 0232.06001)]. The proof uses an induction argument based on the method of ``doubling an element'' which is contained in an earlier paper of \textit{G. Grätzer} [Proc. Am. Math. Soc. 43, 269--271 (1974; Zbl 0292.06003)].
scientific article; zbMATH DE number 5652299