The "walrus operator", as it is known in some circles.

@NilsMatthes thanks!

When first learning about universal covers and Galois correspondence of topological spaces, I remember feeling that conditions like "path-connected and semi-locally simply connected" were rather subtle.

@levborisov I believe the answer is yes and that "arithmetic toric varieties" is a relevant term.

Do you know where I can find Pottharst's notes? I've been trying to track them down but haven't found them.

Is there somewhere else to find these notes? What about the writing by Pottharst mentioned at ncatlab.org/nlab/show/logarithmic+geometry?

@ANonnyMouse I would also be very interested to learn more about the connection between matroids and tropical varieties

@PerAlexandersson Nice!

More parameters, meaning "catalytic variables"?

@HarryGindi I got a sense that something like this was the case, from the first reference, but I'm not used to thinking about lattices and partial orders, so I didn't gain much from this. I know a la here this captures most of Zariski topology, but somehow that doesn't console me.

Thanks. Still need to think a bit more about your remarks, but it's helpful. It doesn't change your conclusion, but wouldn't $S_{\{p\}}$ invert $p$ and the numbers congruent to $1$ mod $p$? I also suspected they were commutative, but couldn't prove, and had reservations relating to regular sequences (I'm editing to add).

Of course. Thank you.