Wrong explanation in chap 2, section 3.2.1.1 Relating Orderings

[fraterenz]

Thank you very much for making this book available, it's very interesting and very well written.

I spotted a mistake (I think?) in the explanation in the relating orderings chapter.
I think for the pushforward orderings we need f to be bijective, but the reason is simply that we need to map each input to the output space, otherwise orderings won't be defined in Y.

So this is correct I think: "If f is not also surjective, and hence bijective, then we cannot pushforward orderings on the input space to orderings on the output space." But the explanation is wrong (see snippet):

quarto_chapters/2_mathematical_spaces/spaces.qmd

Lines 2007 to 2020 in 8fd23d2

	orderings on the output space. The problem is that if any pair
	of output elements are associated with multiple input elements,
	\begin{align*}
	y_{1} &= f(x_1) = f(x_1')
	\\
	y_{2} &= f(x_2) = f(x_2'),
	\end{align*}
	then we can have conflicting input orderings,
	\begin{align*}
	x_1 &< x_2
	\\
	x_2' &< x_1',
	\end{align*}
	that cannot be combined into a consistent output ordering.

Does it make sense? Thanks!

Francesco