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Re: "twisted" Galois connections?
Philip Wadler writes:
> [----- 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
There is a concept called *Lagois* connections which resembles what
you are looking for.
In addition to the conditions above, it is also required that
f(g(f(x))) = x and
g(f(g(a))) = a
See <http://www.math.ksu.edu/~strecker/lagois.ps> for more details.
Sven-Olof Nystrom