Timeline for Cardinality of the set of functions commuting with $f:X\to X$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 4, 2017 at 13:26 | comment | added | Stefan Mesken | @Joel No, there is no such $f$. See Will's answer. | |
| Jan 4, 2017 at 12:56 | comment | added | Joel Adler | Is it obvious that for $\kappa$ singular, there is $f:\kappa\rightarrow\kappa$ such that $\text{card}(\{g:\kappa\rightarrow\kappa\colon f\circ g=g\circ f\})=\text{cf}(\kappa)$? | |
| Dec 30, 2016 at 16:46 | history | edited | Stefan Mesken | CC BY-SA 3.0 |
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| Dec 30, 2016 at 15:01 | history | edited | Stefan Mesken | CC BY-SA 3.0 |
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| Dec 30, 2016 at 14:57 | comment | added | Stefan Mesken | Also note that if $\operatorname{cf}(\kappa) > \omega$, the approach in Lemma 1 still yields $\operatorname{cf}(\kappa)$ many functions commuting with $f$. | |
| Dec 30, 2016 at 14:55 | history | edited | Stefan Mesken | CC BY-SA 3.0 |
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| Dec 30, 2016 at 14:52 | comment | added | Stefan Mesken | @Pietro To be honest, I didn't read the proof I linked. My claimed proof was purely based on Dominic's comment below that answer. So, until I have more time to investigate, exclude the case $\kappa = \omega$ from my answer. I'll handle this case asap. | |
| Dec 30, 2016 at 14:36 | comment | added | Pietro Majer | As to Lemma 1, shouldn't we treat the case $\{f^n\}$ finite (e.g. $f^{12}=f^7$)? | |
| Dec 30, 2016 at 14:30 | comment | added | Stefan Mesken | I think I can prove that we have at least $\operatorname{cf}(\kappa)$ many functions that commutate with any given $f \colon \kappa \to \kappa$. But unfortunately I have to run some other errands first. | |
| Dec 30, 2016 at 14:27 | history | answered | Stefan Mesken | CC BY-SA 3.0 |