The spectrum of a finite pseudocomplemented lattice.

*(English)*Zbl 1209.06004Let \(L\) 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)].

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)].

Reviewer: G. Eigenthaler (Wien)

##### MSC:

06D15 | Pseudocomplemented lattices |

06A11 | Algebraic aspects of posets |

06E05 | Structure theory of Boolean algebras |

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\textit{G. Grätzer} et al., Algebra Univers. 61, No. 3--4, 407--411 (2009; Zbl 1209.06004)

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##### References:

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