Preference relation Let $X= \{(a,b)|a,b\in \mathbb R\}$. Suppose that we have a weak preference relation $R$ (its strict part is denoted by $P$ defined on $X$. Assume that $P$ is coordinate-wise strictly monotonic increasing, that is,
if $a>c$, then $(a,b)P(c,b)$ for all $b$, and
if $b>d$, then $(a,b)P(a,d)$ for all $a$.
Now I want to find an example for $R$ with the properties above that can be represented by a value function, and a counter example that can not be represented by a value function.
I was wondering if someone could help me? 

For a positive example, define $(a,b) R (c,d)$ if and only if $a+b>c+d$. The representing function is $v(x,y) = x + y$ which is of course component-wise strictly monotonic.
EDIT
For a counterexample, use the lexicographic preference relation, defined by $(a,b) R (c,d)$ if and only if $a > c$ or [$a=c$ and $b>d$]. The proof that the lexicographic preference relation admits no value function is well-known; see f.i. https://economics.stackexchange.com/questions/6889/lexicographic-preference-relation-cannot-be-represented-by-a-utility-function
