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Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega_1$, $\omega_2$, we have $$ X(\omega_1)\le X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\omega_2). $$

Edit: As Louigi Addario-Berry's answer below shows, it may be better to consider the following variation: $$ X(\omega_1)< X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\omega_2). $$

Does this property have a name?
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Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega_1$, $\omega_2$, we have $$ X(\omega_1)\le X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\omega_2). $$

Does this property have a name?

And:

To what extent does it follow that $X$ and $Y$ are monotonic functions $f(Z)$, $g(Z)$, respectively, of some real-valued random variable $Z$? Any particularly nice/interesting counterexamples?

(This is more out of curiosity than any direct research-value.)
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Strongly correlated? Terminology question

Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega_1$, $\omega_2$, we have $$ X(\omega_1)\le X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\omega_2). $$

Does this property have a name?

And:

To what extent does it follow that $X$ and $Y$ are monotonic functions $f(Z)$, $g(Z)$, respectively, of some real-valued random variable $Z$? Any particularly nice/interesting counterexamples?

(This is more out of curiosity than any direct research-value.)