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if I have a partially ordered set $P$, and I have a function $f: P \times P \to \mathbb{R}$ that is monotone over the first and antitone over the second argument, i.e. for any $a,b,c \in P$

$a ≤ b \implies f(a,c) ≤ f(b,c)$

and

$a ≤ b \implies f(c,a) ≥ f(c,b)$,

is there anything I can say about the form this equation has to take? For instance, how I could split this up into a function of some unary function, i.e. $f(a,b) \equiv g(h(a),h(b))$?

Thanks a lot, Paul

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  • $\begingroup$ You can consider $f$ also as a monotone function $f:P\times P^{\mathrm{op}}\to\mathbb{R}$. A nice example of such an $f$ is the hom-functor $(a,b)\mapsto 1$ if $a\geq b$ and $(a,b)\mapsto 0$ otherwise. $\endgroup$ Oct 22, 2015 at 14:47
  • $\begingroup$ See also ncatlab.org/nlab/show/profunctor $\endgroup$ Oct 22, 2015 at 14:48

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