# 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 $P\lx@stackrel{{f}}{{\multimap}}Q$, 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.

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, , 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 2013-03-22 15:08:15 Last modified on 2013-03-22 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