Abstract
For every integer n with \(n \geqslant 6\), we prove that the Boolean dimension of a poset consisting of all the subsets of \(\{1,\dots ,n\}\) equipped with the inclusion relation is strictly less than n.
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M. Briański, J. Hodor, H. La, and P. Micek are partially supported by a Polish National Science Center grant (BEETHOVEN; UMO-2018/31/G/ST1/03718).
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Appendix A The Code to Verify Lemma 6
Appendix A The Code to Verify Lemma 6
We provide a short Python code that verifies whether the five linear orders in Table 1 and the function \(\phi \) defined in the proof of Lemma 6 form a Boolean realizer of \(\mathcal {B}_{6}\). Beneath the code, we give the formatted linear orders so that they can be directly inputted into the script for verification.
The input consists of 5 lines. Each represents one of the linear orders in Table 1. The subsets of [6] are converted to corresponding decimal integers based on their binary representations. For example, the number 13 corresponds to the set \(\{ 1, 3, 4 \}\), since the binary representation of 13 on 6 bits is \(001101_2\).
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Briański, M., Hodor, J., La, H. et al. Boolean Dimension of a Boolean Lattice. Order (2024). https://doi.org/10.1007/s11083-024-09666-w
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DOI: https://doi.org/10.1007/s11083-024-09666-w