Timeline for Progress on determining which partial orders embed into the rationals
Current License: CC BY-SA 4.0
25 events
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| Sep 9, 2023 at 22:16 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 9, 2023 at 21:58 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 9, 2023 at 21:31 | comment | added | Holo | @bof yeah, I think it is fine | |
| Sep 9, 2023 at 18:36 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 9, 2023 at 18:04 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 9, 2023 at 17:58 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 9, 2023 at 17:29 | comment | added | Holo | @bof the original reference is in French I believe (I think it is this), but you can find a mention of this fact in "Antichain Decompositions of a Partially Ordered Set" | |
| Sep 9, 2023 at 15:51 | comment | added | Holo | I noticed I made a mistake in my previous comment, the correct function is $f:P→T_2(κ)$ with $f\in 2^{ι(x){\boldsymbol{+1}}}$ | |
| Sep 9, 2023 at 10:46 | comment | added | Holo | The last Proposition is a theorem by Kurepa, who proved the more general form statement: For a cardinal $κ$ let $T_2(κ)=(2^{<κ}, <_{lex})$, if $P$ is a union of $κ$-many anti-chains, then it is embeddable into $T_2(κ)$. Let $(A_α\mid α<κ)$ be some well-ordering of an antichain decomposition (as we may assume they are disjoint), and let $ι(x)$ to be the unique $α$ with $x∈A_α$. Define the witness $f:P→T_2(κ)$ by $f(x)∈2^{ι(x)}$ with $f(x)_α=1$ iff there exists $y∈A_α$ with $y≤x$ | |
| Sep 8, 2023 at 22:57 | vote | accept | C7X | ||
| Sep 8, 2023 at 19:00 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 7:20 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 5:05 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 4:43 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 4:24 | comment | added | C7X | +1, clever poset! About the last paragraph and the comment on the original question, is it shown over ZFC+MA that $P$ is embeddable in $\mathbb Q$ iff $P$ is the union of countably many antichains? | |
| Sep 8, 2023 at 3:19 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 3:18 | history | rollback | bof |
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| Sep 8, 2023 at 3:17 | history | rollback | bof |
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| Sep 8, 2023 at 3:16 | history | rollback | bof |
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| Sep 8, 2023 at 2:53 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 2:46 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 2:34 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 8, 2023 at 0:13 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 7, 2023 at 23:56 | history | edited | bof | CC BY-SA 4.0 |
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| Sep 7, 2023 at 23:48 | history | answered | bof | CC BY-SA 4.0 |