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If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y are partially ordered sets.

So monotonicity is sufficient for preserving extrema. Are necessary and sufficient conditions known for $g$ to preserve minima? If so does anyone know a good reference?

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    $\begingroup$ I think you mean $f: X \to Y$ and $g: Y \to Z$, where $Y$ and $Z$ are partially ordered sets: otherwise $\text{argmin}$ has no meaning. $\endgroup$
    – Robert Israel
    Commented Sep 7, 2018 at 15:32
  • $\begingroup$ Monotonicity is not sufficient, If $g$ is monotone decreasing then the minima is not preserved. Consider $X = [0, 1], f(x) = x, g(x) = 1-x$ $\endgroup$
    – Pushpendre
    Commented Sep 7, 2018 at 15:39
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    $\begingroup$ Actually I think you want it totally ordered. In a partially ordered set, there are two possible meanings of argmin (you may require $f(x)$ to be a minimal element of $f(X)$ or a least element of $f(X)$), and neither of those is preserved by (strictly) monotone increasing functions $g$. The problem is that you could have $g(f(a)) < g(f(b))$ where $f(a)$ and $f(b)$ are incomparable. $\endgroup$
    – Robert Israel
    Commented Sep 7, 2018 at 21:04

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