Maths - Galois Connection

A Galois Connection is pair of adjoint functors between two categories that arise from partially ordered sets.

let:

P = <P, ≤ >

Q = <Q, ∈>

and:

f*: PQ

f*: Q P

for all p∈P and q∈Q

f*(p)∈q iff p≤f*(q)

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

galios connection

Example

preordered set 1

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')

  So is there a mapping (f*) back from Q to P that preserves as much structure as possible in some unique way?
example

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.

example

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).

example

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*.


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