Timeline for Whether an isotone bijection from a power set lattice to another sends singletons to singletons
Current License: CC BY-SA 4.0
25 events
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| Aug 4 at 13:37 | comment | added | Joel David Hamkins | Regarding the injective isotone map question, there can be no isotone injective map from $P(\kappa)$ to $P(\lambda)$, when $\kappa>\lambda$, since the former has chains of order type $\kappa$, but the latter does not. In particular, this also answers my question about almost-disjoint coding---we cannot find coding sets of subsets $A\subseteq\omega_1$ with $a\subseteq\omega$ that respect inclusion. | |
| S Aug 4 at 12:20 | vote | accept | Salvo Tringali | ||
| Aug 4 at 9:27 | comment | added | Salvo Tringali | Just noted a related thread on Math SE: math.stackexchange.com/questions/2897680 | |
| Aug 4 at 8:03 | history | became hot network question | |||
| Aug 4 at 8:03 | answer | added | David Gao | timeline score: 13 | |
| Aug 4 at 7:07 | comment | added | Emil Jeřábek | @MihaHabič No, there is not. This is also shown in my answer. | |
| Aug 4 at 6:34 | vote | accept | Salvo Tringali | ||
| S Aug 4 at 12:20 | |||||
| Aug 4 at 6:02 | comment | added | Salvo Tringali | Concerning my last comment, $h$ sends singletons to singletons and is continuous (wrt the order topology induced by set inclusion). So, $h$ is indeed induced by a permutation of $S$. | |
| Aug 4 at 5:52 | comment | added | Salvo Tringali | @MihaHabič If $f$ and $g$ are permutations of $S$, then the assignment $h: X \mapsto S \setminus g[S \setminus f[X]]$ is an order isomorphism of the power set lattice of $S$. Is $h$ induced by a permutation of $S$? | |
| Aug 4 at 5:44 | answer | added | Emil Jeřábek | timeline score: 13 | |
| Aug 4 at 2:05 | comment | added | Joel David Hamkins | I guess almost disjoint coding is always isotone in the upward direction, since a larger coding set will always code a larger subset of $\omega_1$, wrt inclusion. But that direction isn't injective anyway. I was thinking of the other direction. | |
| Aug 3 at 23:14 | comment | added | Miha Habič | This isn't exactly what you asked, but is related: is there an isotone bijection between powersets which is not induced by a map between the underlying sets (that is, just taking pointwise images)? | |
| Aug 3 at 22:31 | comment | added | Joel David Hamkins | I might ask the question about isotone almost disjoint coding as a separate we question tomorrow, inspired by this question, if that is alright. | |
| Aug 3 at 18:14 | comment | added | Joel David Hamkins | My comments on the almost disjoint coding would be aimed at constructing a counterexample. | |
| Aug 3 at 18:12 | comment | added | Salvo Tringali | Sorry, I read hastily and linked the adjective 'natural' to 'isotone injection' (rather than to 'instance'). I'll think about your question, I don't know the answer off the top of my head. | |
| Aug 3 at 18:07 | comment | added | Joel David Hamkins | But my remarks on almost disjoint coding do refer to the set-theoretic details. | |
| Aug 3 at 18:06 | comment | added | Joel David Hamkins | I just meant an injective function from P(S) to P(T) that is isotone. Does this imply that S injects into T? It seems a related and perhaps more tractable version of your question. | |
| Aug 3 at 17:50 | comment | added | Salvo Tringali | @JoelDavidHamkins I don't even know what 'natural isotone injection' means in this context. Is it something related to the independence of CH from ZFC? I'm not familiar with the details of any proof of Cohen's result. | |
| Aug 3 at 17:43 | comment | added | Joel David Hamkins | In regard to my first comment, I guess it's obvious that there are isotone injective self-embeddings $P(S)\to P(S)$ that don't take singletons to singletons. But one could still ask whether, if there is an isotone injection $P(S)\to P(T)$, does this imply $S$ injects into $T$? | |
| Aug 3 at 17:39 | comment | added | Joel David Hamkins | One common way to show the consistency of Luzin's axiom $2^\omega=2^{\omega_1}$, which is an instance of bijective power sets without bijective underlying sets, is to do almost-disjoint coding from MA+$\neg$CH. In this coding, every subset $A\subseteq\omega_1$ is coded by a set $a\subseteq\omega$, which provides the converse injection. I wonder whether we can find coding sets that make an isotone injection? That would seem natural, based on the how the coding works, but it isn't clear to me how to achieve it. | |
| Aug 3 at 17:37 | comment | added | Joel David Hamkins | Do you know the answer for the natural isotone injection instance of your question? | |
| Aug 3 at 17:30 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Aug 3 at 8:26 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Aug 3 at 8:12 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Aug 3 at 8:04 | history | asked | Salvo Tringali | CC BY-SA 4.0 |