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