Details
| Original language | English |
|---|---|
| Pages (from-to) | 637-645 |
| Number of pages | 9 |
| Journal | Graphs and combinatorics |
| Volume | 17 |
| Issue number | 4 |
| Publication status | Published - Dec 2001 |
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 pn and qn 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 ≤ pn ≤ qn ≤ pn + l(n) + l(l(n)), where l(n) = min{k ⊂ ℕ | n ≤ 2k}. For the smallest size bn of spaces S so that Sub S contains a principal filter isomorphic to the power set script P sign(n), we show n + l(n) - 1 ≤ bn ≤ n + l(n) + l(l(n)) + 2. Since pn ≤ bn, we thus improve recent results of McCluskey and McMaster who obtained pn ≤ n2. For infinite n, we obtain the equation bn = pn = qn = n.
Keywords
- Boolean, Embedding, Poset, Quasiordered set, Representation, Space, Subspace
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Graphs and combinatorics, Vol. 17, No. 4, 12.2001, p. 637-645.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Embedding structures
AU - Erné, Marcel
AU - Reinhold, Jürgen
PY - 2001/12
Y1 - 2001/12
N2 - 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 pn and qn 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 ≤ pn ≤ qn ≤ pn + l(n) + l(l(n)), where l(n) = min{k ⊂ ℕ | n ≤ 2k}. For the smallest size bn of spaces S so that Sub S contains a principal filter isomorphic to the power set script P sign(n), we show n + l(n) - 1 ≤ bn ≤ n + l(n) + l(l(n)) + 2. Since pn ≤ bn, we thus improve recent results of McCluskey and McMaster who obtained pn ≤ n2. For infinite n, we obtain the equation bn = pn = qn = n.
AB - 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 pn and qn 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 ≤ pn ≤ qn ≤ pn + l(n) + l(l(n)), where l(n) = min{k ⊂ ℕ | n ≤ 2k}. For the smallest size bn of spaces S so that Sub S contains a principal filter isomorphic to the power set script P sign(n), we show n + l(n) - 1 ≤ bn ≤ n + l(n) + l(l(n)) + 2. Since pn ≤ bn, we thus improve recent results of McCluskey and McMaster who obtained pn ≤ n2. For infinite n, we obtain the equation bn = pn = qn = n.
KW - Boolean
KW - Embedding
KW - Poset
KW - Quasiordered set
KW - Representation
KW - Space
KW - Subspace
UR - http://www.scopus.com/inward/record.url?scp=19544367126&partnerID=8YFLogxK
U2 - 10.1007/PL00007255
DO - 10.1007/PL00007255
M3 - Article
AN - SCOPUS:19544367126
VL - 17
SP - 637
EP - 645
JO - Graphs and combinatorics
JF - Graphs and combinatorics
SN - 0911-0119
IS - 4
ER -