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 onetoone 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 $\mathrm{Gal}(L/F)$ such that the bijection is inclusionreversing:
$$\mathrm{Gal}(L/F)\supseteq H\supseteq \u27e8e\u27e9\mathit{\hspace{1em}}\text{iff}\mathit{\hspace{1em}}F\subseteq {L}^{H}\subseteq L,\text{and}$$ 
$$F\subseteq K\subseteq L\mathit{\hspace{1em}}\text{iff}\mathit{\hspace{1em}}\mathrm{Gal}(L/F)\supseteq \mathrm{Gal}(L/K)\supseteq \u27e8e\u27e9.$$ 
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 orderedsets:
Definition. Let $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$ be two posets. A Galois connection between $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$ is a pair of functions $f:=({f}^{*},{f}_{*})$ with ${f}^{*}:P\to Q$ and ${f}_{*}:Q\to P$, such that, for all $p\in P$ and $q\in Q$, we have
$${f}^{*}(p){\le}_{Q}q\mathit{\hspace{1em}}\text{iff}\mathit{\hspace{1em}}p{\le}_{P}{f}_{*}(q).$$ 
We denote a Galois connection between $P$ and $Q$ by $P\stackrel{f}{\u22b8}Q$, or simply $P\u22b8Q$.
If we define ${\le}_{P}^{\prime}$ on $P$ by $a{\le}_{P}^{\prime}b$ iff $b{\le}_{P}a$, and define ${\le}_{Q}^{\prime}$ on $Q$ by $c{\le}_{Q}^{\prime}d$ iff $d{\le}_{Q}c$, then $(P,{\le}_{P}^{\prime})$ and $(Q,{\le}_{Q}^{\prime})$ are posets, (the duals of $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$). The existence of a Galois connection between $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$ is the same as the existence of a Galois connection between $(Q,{\le}_{Q}^{\prime})$ and $(P,{\le}_{P}^{\prime})$. 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.
Since ${f}^{*}(p){\le}_{Q}{f}^{*}(p)$ for all $p\in P$, then by definition, $p{\le}_{P}{f}_{*}{f}^{*}(p)$. Alternatively, we can write
${1}_{P}{\le}_{P}{f}_{*}{f}^{*},$ (1) where ${1}_{P}$ stands for the identity map on $P$. Similarly, if ${1}_{Q}$ is the identity map on $Q$, then
${f}^{*}{f}_{*}{\le}_{Q}{1}_{Q}.$ (2) 
2.
Suppose $a{\le}_{P}b$. Since $b{\le}_{P}{f}_{*}{f}^{*}(b)$ by the remark above, $a{\le}_{P}{f}_{*}{f}^{*}(b)$ and so by definition, ${f}^{*}(a){\le}_{Q}{f}^{*}(b)$. This shows that ${f}^{*}$ is monotone. Likewise, ${f}_{*}$ is also monotone.

3.
Now back to Inequality (1), ${1}_{P}{\le}_{P}{f}_{*}{f}^{*}$ in the first remark. Applying the second remark, we obtain
${f}^{*}{\le}_{Q}{f}^{*}{f}_{*}{f}^{*}.$ (3) Next, according to Inequality (2), ${f}^{*}{f}_{*}(q){\le}_{Q}q$ for any $q\in Q$, it is true, in particular, when $q={f}^{*}(p)$. Therefore, we also have
${f}^{*}{f}_{*}{f}^{*}{\le}_{Q}{f}^{*}.$ (4) Putting Inequalities (3) and (4) together we have
${f}^{*}{f}_{*}{f}^{*}={f}^{*}.$ (5) Similarly,
${f}_{*}{f}^{*}{f}_{*}={f}_{*}.$ (6) 
4.
If $(f,g)$ and $(f,h)$ are Galois connections between $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$, then $g=h$. To see this, observe that $p{\le}_{P}g(q)$ iff $f(p){\le}_{Q}q$ iff $p{\le}_{P}h(q)$, for any $p\in P$ and $q\in Q$. In particular, setting $p=g(q)$, we get $g(q){\le}_{P}h(q)$ since $g(q){\le}_{P}g(q)$. Similarly, $h(q){\le}_{P}g(q)$, and therefore $g=h$. By a similarly argument^{}, if $(g,f)$ and $(h,f)$ are Galois connections between $(P,{\le}_{P})$ and $(Q,{\le}_{Q})$, 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 finitedimensional Galois extension^{} of a field $F$, and $G:=\mathrm{Gal}(L/F)$ is the Galois group of $L$ over $F$. If we define

a.
$P=\{K\mid K\text{is a field such that}F\subseteq K\subseteq L\},$ with ${\le}_{P}=\subseteq $,

b.
$Q=\{H\mid H\text{is a subgroup of}G\},$ with ${\le}_{Q}=\supseteq $,

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

a.

•
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 settheoretic inclusion. Next, define ${f}^{*}:P\to Q$ by ${f}^{*}(U)=\overline{U}$, the closure^{} of $U$, and ${f}_{*}:Q\to P$ by ${f}_{*}(V)=\mathrm{int}(V)$, 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 ${f}^{*}(p){\le}_{Q}q$ iff ${f}_{*}(q){\le}_{P}p$, 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)
Title  Galois connection 
Canonical name  GaloisConnection 
Date of creation  20130322 15:08:15 
Last modified on  20130322 15:08:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A15 
Synonym  Galois correspondence 
Synonym  Galois connexion 
Related topic  InteriorAxioms 
Related topic  AdjointFunctor 
Defines  upper adjoint 
Defines  lower adjoint 