Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and
$f(a)=\inf\{b : (a,b)\in R\}$
$g(b)=\inf\{a : (a,b)\in R\}$
then what can we call $f$ and $g$? Perhaps there is some standard terminology.
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asked Nov 4, 2014 at 23:40
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2$\begingroup$ Antitone Galois connections? (At least in some cases.) $\endgroup$– François G. DoraisCommented Nov 5, 2014 at 3:52
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$\begingroup$ Note that $\mathbb{N}$ is not complete with respect to the usual ordering $\leq$ because $\mathbb{N}$ does not have a supremum. $\endgroup$– Dominic van der ZypenCommented Nov 5, 2014 at 8:43
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$\begingroup$ Since he only takes infimums, he only needs lower semi-complete. $\endgroup$– Joel David HamkinsCommented Nov 5, 2014 at 12:58
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$\begingroup$ I don't know any standard terminology for this, but I would likely refer to $f$ as the lower boundary of $R$ and $g$ as the left boundary. If the order also had supremums, then there would also be a corresponding upper boundary and right boundary. $\endgroup$– Joel David HamkinsCommented Nov 5, 2014 at 13:02
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$\begingroup$ Thanks @JoelDavidHamkins in general that sounds good (in my case of interest $R$ is upward closed in the product order). $\endgroup$– Bjørn Kjos-HanssenCommented Nov 5, 2014 at 18:37
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1 Answer
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The maps $f,g$ constitute a Galois connection, but you have to take $\leq := \supseteq$ in both cases.
answered Nov 11, 2014 at 7:41