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• CommentRowNumber1.
• CommentAuthortyler bryson
• CommentTimeAug 9th 2016
The present nLab page for Filter (https://ncatlab.org/nlab/show/filter) claims the following:

"If L is a complete join-semilattice, then Filters(L) is a complete lattice."

I don't see how this is true and no reference is given. Can someone shed light on this? After two days with it, I suspect this is a typo...but am not sure.

Much obliged,
Tyler Bryson
• CommentRowNumber2.
• CommentAuthorDexter Chua
• CommentTimeAug 9th 2016

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.

• CommentRowNumber3.
• CommentAuthortyler bryson
• CommentTimeAug 9th 2016

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.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeAug 27th 2016

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.