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Let $(X, \preceq)$ and $(Y, \ll)$ be partially ordered sets. Suppose there are functions $f : X \to Y$ and $g : Y \to X$ such that $$x \preceq g(y) \iff y \ll f(x)$$ for all $x \in X$, $y \in Y$.

Define the closed elements of $X$ to be all elements of the form $g(f(x))$ and the closed elements of $Y$ to be all elements of the form $f(g(y))$. I have to prove that there is an order-reversing bijection between the closed elements of $X$ and the closed elements of $Y$.

I'm really not sure how to do this. I'm thinking the map would have to be something simple but I can't see what it would be. If you just map $g(f(x)) \mapsto f(x)$ then that isn't necessarily injective and also maps onto $f(X)$ which might be larger than the set of closed elements of $Y$.

I feel like I have to use the property of $f$ and $g$ listed above to derive some other relationship between $f$ and $g$ ... maybe something like $f(g(f(x))) = f(x)$, but I'm not sure.

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  • $\begingroup$ are the functions order-reversing? $\endgroup$ – Stefan Hamcke Sep 4 '13 at 16:20
  • $\begingroup$ The functions just satisfy the condition listed on the third line. I don't think that implies they're order-reversing but I'm not totally sure. $\endgroup$ – saurs Sep 4 '13 at 16:26
  • $\begingroup$ If they are antitone (order-reversing), then this is called a Galois connection. $\endgroup$ – Stefan Hamcke Sep 4 '13 at 16:28
  • $\begingroup$ This reminds me the notion of Galois connection en.wikipedia.org/wiki/Galois_connection . $\endgroup$ – user87690 Sep 4 '13 at 16:29
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You can prove that $fgf = f$ and $gfg = g$. $f(x) ≤ f(x)$ implies $x ≤ gf(x)$, the same for $fg$. Also $x ≤ x' ≤ gf(x')$ implies $f(x) ≥ f(x')$ so in fact the assumption gives that both $f$ and $g$ are order-reversing and they form a Galois connection or adjoint pair on poset categories. $x ≤ gfgf(x)$ gives $f(x) ≥ fgf(x)$ and $gf(x) ≤ gf(x)$ gives $fgf(x) ≥ f(x)$ so $fgf = f$ and $gfg = g$. Now it follows that $f$ and $g$ restricted to closed elements are desired mutually inverse order-reversing bijections ($fgf = f$ implies $gfgf = gf$ so closed elements are exactly fix-points of $gf$ and the same for $fg$. Again by $fgf = f$ we have that closed elements are exactly images of $f$, $g$.)

Also note that the only asymetric result of the construction is $x ≤ gf(x)$ and $y ≤ fg(y)$. If the assumption was $y ≥ f(x) \iff x ≥ g(y)$ then you would have the same result but $x ≥ gf(x)$ and $y ≥ fg(y)$ would hold. This has to do with what is called “contravariant adjoint situation”. Since both $f$, $g$ are order-reversing, neither of them is right nor left adjoint but both are “adjoints on the right”. For the reversed assumption they would be “adjoints on the left”.

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  • $\begingroup$ It should be "... implies $f(x')\le f(x)$" in the third sentence. $\endgroup$ – Stefan Hamcke Sep 4 '13 at 16:46
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    $\begingroup$ You say "$x \leq gfgf(x)$ gives $ f(x) \leq fgf(x)$". Shouldn't that be "$x \leq gfgf(x)$ gives $ fgf(x) \leq f(x)$" or am I missing something? (Thanks for your help by the way) $\endgroup$ – saurs Sep 4 '13 at 16:52
  • $\begingroup$ That's interesting. So far I didn't know that you can drop the monotonicity from the definition of Galois connection and only need that property. $\endgroup$ – Stefan Hamcke Sep 4 '13 at 16:56
  • $\begingroup$ @StefanH., saurs: Yes you are right, I forgot to change the direction of $≤$ when removing $g$ on the right and adding $f$ on the left. $\endgroup$ – user87690 Sep 4 '13 at 16:56
  • $\begingroup$ Similarly $gf(x) \leq gf(x)$ seems to imply that $f(x) \leq fgf(x)$. But the conclusion is the saem either way. $\endgroup$ – saurs Sep 4 '13 at 16:57

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