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Topic: Special Galois connections
Replies: 2   Last Post: Jun 21, 2009 5:57 AM

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galathaea

Posts: 1,646
Registered: 7/2/05
Re: Special Galois connections
Posted: Jun 20, 2009 3:00 PM
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On Jun 20, 8:18 am, Victor Porton <por...@narod.ru> wrote:
> Let A is a set, let functions f: A->A and g: A->A. Let (f; g) is a
> Galois connection where f is the lower adjoint and g is the upper
> adjoint.
>
> I have gf = g. What is known about this special class of Galois
> connections such that gf = f?


the members of A have a pordering associated
right?

for the very special case gf = id
this is just a retract

for gf < id
g and f are sometimes called "projection pairs"

this terminology gives a conceptual framework
that is useful for looking at these things

now look at gf = f
this means that fgf = f^2
but we also know that in general
forAll galois connections
fgf = f

(the proof for this latter fact is to use
c <= gf (c)
and substitute c -> g(c1)
giving g(c1) <= gfg (c1)
and then use the dual identity
fg (c1) <= c1
apply g to both sides
giving gfg (c1) <= g(c1)
so equality follows from the two inequalities)

so f = f^2
and you are looking at an idempotent
like a projection

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar



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