Abstract
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.
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Presented by M. Ploščica.
The research of the first two authors was supported by the NSERC of Canada.
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Grätzer, G., Gunderson, D.S. & Quackenbush, R.W. The spectrum of a finite pseudocomplemented lattice. Algebra Univers. 61, 407 (2009). https://doi.org/10.1007/s00012-009-0027-x
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DOI: https://doi.org/10.1007/s00012-009-0027-x