# 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

```