Timeline for An extension of Stone duality
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2022 at 8:52 | vote | accept | LeopSchl | ||
| Aug 23, 2022 at 15:22 | comment | added | LeopSchl | Oh yes. On the other hand if we restrict to atomic sentences, the construction isn't functorial. | |
| Aug 23, 2022 at 15:14 | comment | added | Alex Kruckman | @LeopSchl I'm not sure. But your proposed partial order is trivial: If $M\leq M'$, then for every $p$, if $M\models p$, then $M'\models p$, and also if $M\models \lnot p$, then $M'\models \lnot p$, so $M = M'$. | |
| Aug 23, 2022 at 15:02 | comment | added | LeopSchl | Thanks! I wonder whether there's another way to make $(L, W)\mapsto W$ into a conservative functor. Here's an idea: we could equip $W$ with a partial order instead of a topology, by setting $M\leq M'$ if and only if for all $L$-sentences $\phi$, $M\models \phi\Rightarrow M'\models \phi$. | |
| Aug 23, 2022 at 13:46 | comment | added | Alex Kruckman | Now there is an interpretation $(L,W)\to (K,V)$ by $p_n\mapsto q_n\land \lnot q_{n-1}$, inducing the homeomorphism $N_n\mapsto M_n$ between $V$ and $W$. But there is no inverse interpretation, since $q_m$ should be true whenever $p_n$ is true for some $n\leq m$ in $\mathbb{Z}$, and this is not expressible by a single sentence. In $\mathsf{Stone}_D$, we can see that the closure of $W$ has a single limit point (none of the $p_n$ are true), while the closure of $V$ has two limit points (all or none of the $q_n$ are true). The interpretation collapses the limit points, so it is not invertible. | |
| Aug 23, 2022 at 13:43 | comment | added | Alex Kruckman | @LeopSchl I have a counterexample to conservativity to $\mathsf{Top}$. Let $L = \{p_n\mid n\in \mathbb{Z}\}$, and let $K = \{q_n\mid n\in \mathbb{Z}\}$. Let $W = \{M_n\mid n\in \omega\}$ where $p_n$ is the only variable true in $M_n$, as above. Topologically, this is a discrete space. Let $V = \{N_n\mid n\in \omega\}$, where $q_m$ is true in $N_n$ if and only if $m\geq n$. Topologically, $V$ is also discrete, since $N_n$ is distinguished by $q_n\land \lnot q_{n-1}$. | |
| Aug 23, 2022 at 13:23 | comment | added | Alex Kruckman | @LeopSchl Certainly the functor to $\mathsf{Stone}_D$ that I suggested is conservative, since it is an equivalence. Your functor to $\mathsf{Top}$ composes this equivalence with the forgetful functor $\mathsf{Stone}_D\to \mathsf{Top}$ which takes the subspace topology on the distinguished dense set. So the question comes down to this: If $f\colon X\to Y$ is a continuous map of stone spaces, which restricts to a homeomorphism between a dense subspace of $X$ and a dense subspace of $Y$, is $f$ itself a homeomorphism? I don't know the answer off the top of my head. | |
| Aug 23, 2022 at 13:13 | comment | added | LeopSchl | Thanks! "Topologically, what's going on is this: [...]" -- so are you saying $\mathrm{PropClass}\to \mathrm{Top}$ is conservative? | |
| Aug 23, 2022 at 13:10 | comment | added | Alex Kruckman | Topologically, what's going on is this: The ambient stone space is $V = \{M_n\mid n\in \omega\}\cup \{M_*\}$, where each $M_n$ is a discrete point and $M_*$ is a limit point. Now the map $M_{n+1}\mapsto M_n$, $M_0\mapsto M_*$, and $M_*\mapsto M_*$ is continuous, but it has no continuous inverse, since it maps the discrete point $M_0$ to the limit point $M_*$. | |
| Aug 23, 2022 at 13:05 | comment | added | Alex Kruckman | The answer is no. Let $L = K = \{p_n\mid n\in \omega\}$. Let $W = \{M_n\mid n\in \omega\}$ where $p_n$ is the only variable true in $M_n$. Let $V = W\cup \{M_*\}$, where none of the $p_n$ are true in $M_*$. There is an interpretation $(L,V)\to (L,W)$ by $p_n\mapsto p_{n+1}$. The induced function $W\to V$ maps $M_{n+1}$ to $M_n$ for $n\geq 0$ and maps $M_0$ to $M_*$, so it is a bijection. But there is no inverse interpretation, since in such an interpretation, $p_0$ should be true when all of the $p_n$ are false, which is not expressible by a single sentence. | |
| Aug 23, 2022 at 12:53 | comment | added | Alex Kruckman | You're asking about the functor $(L,W)\mapsto W$, right? So the question is: if an interpretation between propositional classes induces a bijection between the classes, does it have an inverse interpretation? | |
| Aug 23, 2022 at 9:32 | comment | added | LeopSchl | Thanks! Is the contravariant functor $\mathrm{PropClass}\to \mathrm{Set}$ (note that I replaced $\mathrm{Top}$ by $\mathrm{Set}$) conservative (i.e., isomorphism-reflecting)? | |
| Aug 22, 2022 at 18:48 | history | answered | Alex Kruckman | CC BY-SA 4.0 |