"twisted" Galois connections?

[Philip Wadler <wadler@research.avayalabs.com>]

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

-----------------------------------------------------------------------
Philip Wadler wadler@avaya.com
www.research.avayalabs.com/user/wadler
Avaya Labs, 233 Mount Airy Road, Basking Ridge, NJ 07920 USA
phone +1 908 696 5137 fax +1 908 696 5402
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"When a Mathematical Reasoning can be had it's as great a folly
to make use of any other, as to grope for a thing in the dark,
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