# Re: "twisted" Galois connections?

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On Tuesday, June 17, 2003, at 12:21  PM, Philip Wadler wrote:

> [----- The Types Forum, http://www.cis.upenn.edu/~bcpierce/types -----]
>
> Is anything known about the following variation on a Galois
> connection?
>
> Given domains X and A with partial orders, f:X->A and g:A->X
> constitute a *Galois connection* if the following four conditions
> hold
>
> 	(1)  x <= y   implies   f(x) <= f(y)
> 	(2)  a <= b   implies   g(a) <= g(b)
> 	(3)  x <= g(f(x))
> 	(4)  f(g(a)) <= a
>
> (This is equivalent to saying f(x) <= a iff x <= g(a).)
>
> The same functions constitute a *twisted Galois connection* if
> we have conditions (1)-(3) and also
>
> 	(4')  a <= f(g(a))
>
> Both Galois connections and twisted Galois connections compose.
> If f:X->A, g:A->X and h:A->Z, k:Z->A consitute a (twisted) Galois
> connection, then so do f;h:X->Z, k;g:Z->X.
>
> Is there anything in the literature about twisted Galois connections
> or the corresponding notion of a twisted adjoint, perhaps under
> a different name?  Many thanks,  -- P

Well, I'm not a mathematician, but it seems to me that many of the nice
properties that Galois connections have are lost in the "twisted" one.
For instance, it need not be the case that (f o g) be the identity for
the range of f.  For instance, if I understand you correctly, f(x) =
g(x) = x + 1 constitutes a twisted galois connection between integers &
integers.

john clements
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