Abstract.
For any quasiordered set (`quoset') or topological space S, the set Sub S of all nonempty subquosets or subspaces is quasiordered by embeddability. Given any cardinal number n, denote by p n and q n the smallest size of spaces S such that each poset, respectively, quoset with n points is embeddable in Sub S. For finite n, we prove the inequalities n + 1 ≤p n ≤q n ≤p n + l(n) + l(l(n)), where l(n) = min{k∈ℕ∣n≤2k}. For the smallest size b n of spaces S so that Sub S contains a principal filter isomorphic to the power set ?(n), we show n + l(n) − 1 ≤b n ≤n + l(n) + l(l(n))+2. Since p n ≤b n , we thus improve recent results of McCluskey and McMaster who obtained p n ≤n 2. For infinite n, we obtain the equation b n = p n = q n = n.
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Received: April 19, 1999 Final version received: February 21, 2000
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Erné, M., Reinhold, J. Embedding Structures. Graphs Comb 17, 637–645 (2001). https://doi.org/10.1007/PL00007255
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DOI: https://doi.org/10.1007/PL00007255