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 and its subfield (with appropriate conditions imposed on the extension ), and the subgroups of the Galois group such that the bijection is inclusion-reversing:
Definition. Let and be two posets. A Galois connection between and is a pair of functions with and , such that, for all and , we have
We denote a Galois connection between and by , or simply .
If we define on by iff , and define on by iff , then and are posets, (the duals of and ). The existence of a Galois connection between and is the same as the existence of a Galois connection between and . In short, we say that there is a Galois connection between and if there is a Galois connection between two posets and where and are the underlying sets (of and respectively). With this, we may say without confusion that “a Galois connection exists between and iff a Galois connection exists between and ”.
Since for all , then by definition, . Alternatively, we can write
where stands for the identity map on . Similarly, if is the identity map on , then
Suppose . Since by the remark above, and so by definition, . This shows that is monotone. Likewise, is also monotone.
Now back to Inequality (1), in the first remark. Applying the second remark, we obtain
Next, according to Inequality (2), for any , it is true, in particular, when . Therefore, we also have
Putting Inequalities (3) and (4) together we have
If and are Galois connections between and , then . To see this, observe that iff iff , for any and . In particular, setting , we get since . Similarly, , and therefore . By a similarly argument, if and are Galois connections between and , then . Because of this uniqueness property, in a Galois connection , is called the upper adjoint of and the lower adjoint of .
The most famous example is already mentioned in the first paragraph above: let is a finite-dimensional Galois extension of a field , and is the Galois group of over . If we define
by , and
by , the fixed field of in .
Then, by the fundamental theorem of Galois theory, and are bijections, and is a Galois connection between and .
Let be a topological space. Define be the set of all open subsets of and the set of all closed subsets of . Turn and into posets with the usual set-theoretic inclusion. Next, define by , the closure of , and by , the interior of . Then is a Galois connection between and . Incidentally, those elements fixed by are precisely the regular open sets of , and those fixed by 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 and such that iff , so that the pair are order reversing. In any case, the two definitions are equivalent in that one may go from one definition to another, (simply exchange with , the dual (http://planetmath.org/DualPoset) of ).
|Date of creation||2013-03-22 15:08:15|
|Last modified on||2013-03-22 15:08:15|
|Last modified by||CWoo (3771)|