I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the Galois connection underlying Galois theory in terms of a binary relation between individual elements of the field $E$ and individual elements of the group Gal($E/K$), rather than a binary relation between subfields of $E$ containing $K$ and subgroups of Gal($E/K$). Is there a problem with doing things in an element-by-element way? (The current draft of the article is on the web at http://jamespropp.org/galois.pdf , although you probably don't need to read it to think about this half of my question.)

Another qualm I have is that I suspect that some version of the social network "application" that I present ($K(K(K(S)))=K(S)$, where $K(S) = \ ${$t:s \sim t$ for all $s \in S$}, for some symmetric binary relation $\sim$) occurs in the contest-problem literature or the recreational math literature, and I'd appreciate relevant citations if anyone knows of them.

setsof elements (under the usual partial-ordering by the subset relation). The fixed sets are fields, as you say. – Pace Nielsen Jul 29 '11 at 17:15