Examples of Galois connections? On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end :

But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If you've got a little gadget k sitting in a big gadget K, you get a "Galois group" Gal(K/k) consisting of symmetries of the big gadget that fix everything in the little one.
  But now here's the cool part, which is also very general. Any subgroup of Gal(K/k) gives a gadget containing k and contained in K: namely, the gadget consisting of all the elements of K that are fixed by everything in this subgroup.
  And conversely, any gadget containing k and contained in K gives a subgroup of Gal(K/k): namely, the group consisting of all the symmetries of K that fix every element of this gadget.

Apart from fields, what other "big gadgets" can be described in this way ? And what are the corresponding "little gadgets" ?

This wikipage gives a good list of examples of both monotone and antitone Galois connections.
