Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 4 of 4
A complete join-semilattice is automatically a complete meet-semilattice, so by the first point in the list, $Filters(L)$ is a complete join-semillatice, hence a complete lattice.
Thank you. I should have seen that. I do think the wording on that bullet point could be improved. As it stands, I expect I am not the only person who’ll incorrectly infer that the completeness of the lattice $Filters(L)$ is a consequence of passing to filters and not an easy byproduct of the order on $L$. I may submit an edit.
I think that the point of those bullet points is supposed to be about how $f_*$ has various nice properties when $f$ does. So the first sentence of each is sort of just introductory: if $L$ has some structure, then $Filters(L)$ has some structure, OK, but now here's what's interesting: if $f$ preserves the structure on $L$, then $f_*$ preserves the structure on $Filters(L)$, and that's what we want to talk about! At least, that's the attitude, I think.
1 to 4 of 4