mapping strict order to real function

Question

Let $\succ$ be a strict (either partial or total) order over a set $X$. Is there a mapping function $f:X \rightarrow \mathbb R$ such that $f(x)>f(y)$ if and only if $x\succ y$ or equivalently $(x,y)\in \succ$ .

Answer 1

If your set $X$ is too big then the answer is no. Suppose for example your set $X$ is totally ordered and has cardinality equal to the power set of $\mathbb{R}$. Then any mapping $f:X \to {\mathbb R}$ will map two or more elements $x_1 < x_2$ to the same real number, and so you will have $f(x_1) = f(x_2)$ even though $x_1 < x_2$. If your poset $X$ has cardinality equal to that of ${\mathbb R}$ then the question is more interesting, I'll update if I come up with an answer for that.

If your set is finite, then you can always extend the partial ordering to a total ordering (in linear time in your partial order size), and then just assign real numbers to your elements in order after obtaining the total ordering. See http://en.wikipedia.org/wiki/Linear_extension for how to extend your partial ordering to a total ordering in linear time.