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For where $(a,b)\preceq (c,d): \Leftrightarrow a≤c$ and $b≤d$.

As far as I'm concerned there are none of maxima, minima, largest or smallest elements at all.

Since there can always be a smaller $a, b$ or larger $c, d$.

But I'm not sure if I've understood it correctly. How can I prove that these (don't) exist?

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You are correct that these do not exist. To prove it, just show that if I give you an element $(x,y)$ you can find a smaller and larger one. Then any element I claim is largest or smallest cannot be. When you ask for maxima and minima are you thinking in some local sense? Similarly, given an $(x,y)$ that is claimed to be a local maximum, can you find an element $(x',y')$ that is very close (in the Euclidean distance) to $(x,y)$ and greater?

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    Thanks. By minima and maxima, I thought that referred to sets of multiple largest or smallest elements that all have the same order. – 2017-01-09
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    In your order, no pair of different elements have the same value. You do have pairs that are not comparable, like $(0,1)$ and $(1,0)$, so it is a partial order. Given any collection of points, you can find a point greater than them all – 2017-01-09
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    Any *finite* collection of points. – 2017-01-09