# Galois connection

The notion of a Galois connection has its root in Galois theory. By the fundamental theorem of Galois theory (http://planetmath.org/FundamentalTheoremOfGaloisTheory), there is a one-to-one correspondence between the intermediate fields between a field $L$ and its subfield $F$ (with appropriate conditions imposed on the extension $L/F$), and the subgroups of the Galois group $\operatorname{Gal}(L/F)$ such that the bijection is inclusion-reversing:

 $\operatorname{Gal}(L/F)\supseteq H\supseteq\langle e\rangle\quad\mbox{ iff }% \quad F\subseteq L^{H}\subseteq L,\mbox{ and}$
 $F\subseteq K\subseteq L\quad\mbox{ iff }\quad\operatorname{Gal}(L/F)\supseteq% \operatorname{Gal}(L/K)\supseteq\langle e\rangle.$

If the language of Galois theory is distilled from the above paragraph, what remains reduces to a more basic and general concept in the theory of ordered-sets:

Definition. Let $(P,\leq_{P})$ and $(Q,\leq_{Q})$ be two posets. A Galois connection between $(P,\leq_{P})$ and $(Q,\leq_{Q})$ is a pair of functions $f:=(f^{*},f_{*})$ with $f^{*}\colon P\to Q$ and $f_{*}\colon Q\to P$, such that, for all $p\in P$ and $q\in Q$, we have

 $f^{*}(p)\leq_{Q}q\quad\mbox{ iff }\quad p\leq_{P}f_{*}(q).$

We denote a Galois connection between $P$ and $Q$ by $P\lx@stackrel{{\scriptstyle f}}{{\multimap}}Q$, or simply $P\multimap Q$.

If we define $\leq_{P}^{\prime}$ on $P$ by $a\leq_{P}^{\prime}b$ iff $b\leq_{P}a$, and define $\leq_{Q}^{\prime}$ on $Q$ by $c\leq_{Q}^{\prime}d$ iff $d\leq_{Q}c$, then $(P,\leq_{P}^{\prime})$ and $(Q,\leq_{Q}^{\prime})$ are posets, (the duals of $(P,\leq_{P})$ and $(Q,\leq_{Q})$). The existence of a Galois connection between $(P,\leq_{P})$ and $(Q,\leq_{Q})$ is the same as the existence of a Galois connection between $(Q,\leq_{Q}^{\prime})$ and $(P,\leq_{P}^{\prime})$. In short, we say that there is a Galois connection between $P$ and $Q$ if there is a Galois connection between two posets $S$ and $T$ where $P$ and $Q$ are the underlying sets (of $S$ and $T$ respectively). With this, we may say without confusion that “a Galois connection exists between $P$ and $Q$ iff a Galois connection exists between $Q$ and $P$”.

Remarks.

1. 1.

Since $f^{*}(p)\leq_{Q}f^{*}(p)$ for all $p\in P$, then by definition, $p\leq_{P}f_{*}f^{*}(p)$. Alternatively, we can write

 $\displaystyle 1_{P}\leq_{P}f_{*}f^{*},$ (1)

where $1_{P}$ stands for the identity map on $P$. Similarly, if $1_{Q}$ is the identity map on $Q$, then

 $\displaystyle f^{*}f_{*}\leq_{Q}1_{Q}.$ (2)
2. 2.

Suppose $a\leq_{P}b$. Since $b\leq_{P}f_{*}f^{*}(b)$ by the remark above, $a\leq_{P}f_{*}f^{*}(b)$ and so by definition, $f^{*}(a)\leq_{Q}f^{*}(b)$. This shows that $f^{*}$ is monotone. Likewise, $f_{*}$ is also monotone.

3. 3.

Now back to Inequality (1), $1_{P}\leq_{P}f_{*}f^{*}$ in the first remark. Applying the second remark, we obtain

 $\displaystyle f^{*}\leq_{Q}f^{*}f_{*}f^{*}.$ (3)

Next, according to Inequality (2), $f^{*}f_{*}(q)\leq_{Q}q$ for any $q\in Q$, it is true, in particular, when $q=f^{*}(p)$. Therefore, we also have

 $\displaystyle f^{*}f_{*}f^{*}\leq_{Q}f^{*}.$ (4)

Putting Inequalities (3) and (4) together we have

 $\displaystyle f^{*}f_{*}f^{*}=f^{*}.$ (5)

Similarly,

 $\displaystyle f_{*}f^{*}f_{*}=f_{*}.$ (6)
4. 4.

If $(f,g)$ and $(f,h)$ are Galois connections between $(P,\leq_{P})$ and $(Q,\leq_{Q})$, then $g=h$. To see this, observe that $p\leq_{P}g(q)$ iff $f(p)\leq_{Q}q$ iff $p\leq_{P}h(q)$, for any $p\in P$ and $q\in Q$. In particular, setting $p=g(q)$, we get $g(q)\leq_{P}h(q)$ since $g(q)\leq_{P}g(q)$. Similarly, $h(q)\leq_{P}g(q)$, and therefore $g=h$. By a similarly argument, if $(g,f)$ and $(h,f)$ are Galois connections between $(P,\leq_{P})$ and $(Q,\leq_{Q})$, then $g=h$. Because of this uniqueness property, in a Galois connection $f=(f^{*},f_{*})$, $f^{*}$ is called the upper adjoint of $f_{*}$ and $f_{*}$ the lower adjoint of $f^{*}$.

Examples.

• The most famous example is already mentioned in the first paragraph above: let $L$ is a finite-dimensional Galois extension of a field $F$, and $G:=\operatorname{Gal}(L/F)$ is the Galois group of $L$ over $F$. If we define

• a.

$P=\{K\mid K\mbox{ is a field such that }F\subseteq K\subseteq L\},$ with $\leq_{P}=\subseteq$,

• b.

$Q=\{H\mid H\mbox{ is a subgroup of }G\},$ with $\leq_{Q}=\supseteq$,

• c.

$f^{*}:P\to Q$ by $f^{*}(K)=\operatorname{Gal}(L/K)$, and

• d.

$f_{*}:Q\to P$ by $f_{*}(H)=L^{H}$, the fixed field of $H$ in $L$.

Then, by the fundamental theorem of Galois theory, $f^{*}$ and $f_{*}$ are bijections, and $(f^{*},f_{*})$ is a Galois connection between $P$ and $Q$.

• Let $X$ be a topological space. Define $P$ be the set of all open subsets of $X$ and $Q$ the set of all closed subsets of $X$. Turn $P$ and $Q$ into posets with the usual set-theoretic inclusion. Next, define $f^{*}:P\to Q$ by $f^{*}(U)=\overline{U}$, the closure of $U$, and $f_{*}:Q\to P$ by $f_{*}(V)=\operatorname{int}(V)$, the interior of $V$. Then $(f^{*},f_{*})$ is a Galois connection between $P$ and $Q$. Incidentally, those elements fixed by $f_{*}f^{*}$ are precisely the regular open sets of $X$, and those fixed by $f^{*}f_{*}$ are the regular closed sets.

Remark. The pair of functions in a Galois connection are order preserving as shown above. One may also define a Galois connection as a pair of maps $f^{*}:P\to Q$ and $f_{*}:Q\to P$ such that $f^{*}(p)\leq_{Q}q$ iff $f_{*}(q)\leq_{P}p$, so that the pair $f^{*},f_{*}$ are order reversing. In any case, the two definitions are equivalent in that one may go from one definition to another, (simply exchange $Q$ with $Q^{\partial}$, the dual (http://planetmath.org/DualPoset) of $Q$).

## References

• 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
• 2 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)