Galois Connection

In this article, we discuss the (absctract) notion of Galois connection. Let $\CF$ and $\CG$ be two partially ordered sets. Let $f: \CF \to \CG$ and $g: \CG \to \CF$ be maps.

Definition (Galois connection)

We say $f$ and $g$ establish a Galois connection between the two sets if the following take place:

$g(f(H)) \supseteq H$ and $f(g(K))\supseteq K$ for all $H \in \CF$ and $K \in \CG$.
If $H_1 \subseteq H_2$, then $f(H_1) \supseteq f(H_2)$.
If $K_1 \subseteq K_2$, then $g(K_1) \supseteq g(K_2)$.

$$ \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Gal}{Gal} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} \DeclarePairedDelimiter{\norm}{\lVert}{\rVert} $$

Galois connection arise frequently in mathematics. For example, let $F \subseteq E$ be field extension, then the map between the set of intermediate fields between $F$ and $E$, and the set of subgroups of $\Gal(E/F)$, establish a galois connection. This is the well-known galois theory c.f. [1, page 275] .

Another typical example is the map between the set of ideals of $k[T_1,\cdots,T_n]$ and the set of subsets of affine space of dimension $n$ (or set of subsets of the spectrum of $k[T_1,\cdots,T_n]$). This is the well-known Hilbert’s Nullstellensatz.

Proposition (Bijection induced from galois connection)

$f$ and $g$ define inverse bijections between the subsets $\CF_0 \subseteq \CF$ and $\CG_0 \subseteq \CG$ defined by $\CF_0 := \{g(K): K \in \CG\}, \CG_0 := \{f(H): H \in \CF\}$. Also, $$ \CF_0 = \{H \in \CF: g(f(H)) = H\} \qquad \CG_0 = \{K \in \CG: f(g(K)) = K\}. $$

Proof. Clearly $f(\CF_0) \subseteq f(\CF) = \CG_0$, so $f$ defines a map $\CF_0 \to \CG_0$, the same is true of $g$.

Let $H \in \CF_0$, on the one hand, we have $g(f(H)) \supseteq H$ by definition. On the other hand, we can write $H = g(K)$ for some $K \in \CG$ since $H \in \CF_0$, and so $f(H) = f(g(K)) \supseteq K$ $\implies$ $g(f(H)) \subseteq g(K) = H$. Hence $g\circ f = \id$ on $\CF_0$. We can similarly show that $f \circ g = \id$ on $\CG_0$.

If $H \in \CF_0$, we have already proved that $g(f(H)) = H$. Conversely, if $H \in \CF$ such that $H = g(f(H))$, then $H = g(K) \in \CF_0$ where $K = f(H) \in \CG$. $\square$

The elements of $\CF_0$ and $\CG_0$ are called the closed elements of $\CF$ and $\CG$. The maps $g\circ f: \CF \to \CF$ and $f\circ g: \CG \to \CG$ are the closure operators on $\CF$ and $\CG$.

References

1: I. Martin Isaacs. Algebra: A Graduate Course. GSM 100.