Monotone map on a Preset Related to my previous question I ran across this in the same book and thought I would try to get some clarity.
If $R$ and $S$ are both Presets with relation $\le$ then the function $f$ is a monotone map from $R$ to $S$ ($R \xrightarrow{f} S$) if:
$$\forall x,y \in R: \ x \le y \implies f(x) \le f(y)$$
This is all well and good, but obviously the first $\le$ is on the $R$ Preset and the second $\le$ is on the $S$ Preset.  Does this definition have any implication on the meaning of the two relations $\le_R$ and $\le_S$?  Do they have to be the 'same' relation (whatever that may mean)?
My guess is that it has no such implication but that leaves me wondering what practical use this really has since it would seem to come down to how clever you are at defining your relations in the two sets.
 A: Yes, they can be different in nature:
 Consider e.g. the identity map for $f$ from $R=(\Bbb N,\,|\,)$ to $S=(\Bbb N,\le)$. 
A: In mathematics, when you study sets with some extra structure/operation (for ex. multiplication in groups), you are also interested in studying functions that "respect" that structure (for ex. group-homomorphisms). With presets (also known as prosets=PRe-Ordered SETS) the structure/operation is the pre-order $\le$.
So with groups you are interested in functions $f$ such that:
$$\forall x,y \in G: f (x * y) = f(x) * f(y)$$
and with presets you study functions $f:R \to S$ such that
$$\forall x,y \in R: \ x \le_R y \implies f(x) \le_S f(y)$$
Do you see the similitude?
In a short while you will read that presets are a very "skinny" or "thin" kind of category and that monotone functions are functors and things will be even clearer.
At that point you will interpret $R$ and $S$ as thin categories and $\le_R$ and $\le_S$ as arrows in the respective categories. The monotone function $f:R \to S$ will be interpreted as a functor and you will be able to literally write:
$$\forall x,y \in R: f( x \le_R y) = f(x) \le_S f(y)$$
thereby fully mimicking the algebraic (for ex. group theoretic) formula given above.
