I remind that I am not a professional mathematician. Nevertheless I have written research monograph “Algebraic General Topology. Volume 1”.
Yesterday I have asked on MathOverflow how to characterize a poset of all filters on a set.
From the answer: the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.
Honestly? I don’t understand it. I need to learn.
The above characterization of filters may be useful for study of filters (and reloids) themselves. Earlier I considered frames and locales as a possible special case of pointfree funcoids (however I have not yet demonstrated any interrelations between pointfree funcoids and frames/locales). And this was not a first priority for me to consider frames, locales, and their relationships with pointfree funcoids. But now frames and locales become more important for me: It is a tool to study filters themselves.
Some time ago (maybe a year ago) I bought the book Peter T. Johnstone “Stone Spaces”. I did a few attempt to start to understand what it is about, but failed. Now (after I finish checking of errors in my book) I am going to seriously read “Stone Spaces”.