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The spectrum of a finite pseudocomplemented lattice. (English) Zbl 1209.06004
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 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 G. Grätzer [Proc. Am. Math. Soc. 43, 269–271 (1974; Zbl 0292.06003)].

MSC:
06D15 Pseudocomplemented lattices
06A11 Algebraic aspects of posets
06E05 Structure theory of Boolean algebras
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References:
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