Galois connections

Question

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.

Answer 1

You are right: there is nothing to worry about. What you are describing is quite commonplace with Galois connections. In fact, "elementwise" binary relations are arguably the number one source of Galois connections.

Let $R \subseteq X \times Y$ be any binary relation. Then there is an induced Galois connection, a pair of contravariant (i.e., order-reversing) maps

$$P(X) \stackrel{-\backslash R}{\to} P(Y): S \mapsto \{y \in Y: \forall_{x \in X} \ \ x \in S \Rightarrow R(x, y)\}$$

$$P(Y) \stackrel{R/-}{\to} P(X): T \mapsto \{x \in X: \forall_{y \in Y} \ \ y \in T \Rightarrow R(x, y)\}$$

for which

$$S \subseteq R/T \qquad \text{iff} \qquad S \times T \subseteq R \qquad \text{iff} \qquad T \subseteq S\backslash R$$

One may compose these operations in either direction,

$$S \mapsto R/(S \backslash R), \qquad T \mapsto (R/T)\backslash R$$

to get closure operations satisfying the standard axioms, e.g., $S \subseteq S'$ implies $\bar{S} \subseteq \bar{S'}$ and

$$S \subseteq \bar{S}, \qquad \bar{\bar{S}} = \bar{S}$$

for all $S \in P(X)$. This may be proven just by taking advantage of the three-way equivalence above together with the contravariance.

Given a Galois extension $K/k$ with Galois group $G$, the Galois connection of Galois theory is induced by the subset $R \subseteq G \times K$ defined by

$$\{(\sigma, x) \in G \times K: \sigma(x) = x\}$$

and there is nothing wrong with considering the induced Galois connection on power sets rather than the posets of subgroups and intermediate subfields. The closed elements in this case are the subgroups and intermediate subfields! This statement is more or less the substance of the fundamental theorem of Galois theory. But my real point here is the utter generality of these sorts of Galois connections.