Maths - Galois Connection

let:

P = <P, ≤ >

Q = <Q,>

and:

f_*: P->Q

f^*: Q->P

for all pP and qQ

f_*(p)q iff p≤f^*(q)

The pair <f_*,f^*> form a Galois connection between P and Q.

Here we look at a perordered set ≤(red) and a mapping to another perordered set (blue). The morphism between them (green) preserves the ordering. That is:

if p ≤ p' then f_*(p) ≤ f_*(p')

Here is a first attempt, as we can see we can compose these morphisms and still preseve the order structure
:if a ≤ a' then f^*f_*(p) ≤f^*f_*(p'). However, it does not do this in a unique way, there are many mappings for f^* that could have done this.

So lets add another condition:

f^*f_*(p) ≤ p

This condition is not met for our first choice of f^* (on the diagram on the left this is interpreted as reversing one of the red arrows).

However, given the morphism f_* then there is a way to choose a way to choose the morphism f^* so that the order is preserved relative to the original sequence.

For instance, if the number the elements of 'P' with ascending values. We then map these values onto 'Q' using the morphism 'f_*'. We can then determine f^* by mapping the highest value for each element in 'Q' to the corresponding number in 'P'.

f^* is then the adjoint of f_*.