Functional analysis, fixed point theory

Question

In the theory of Riesz spaces, I am unable to understand a point.

If $aRb$ and $cRd$ then it is necessarily true that $a+cRb+d$, where $R$ shows partial order relation. If this is not true, please provide a counterexample.

Thanks in advance

Answer 1

I will replace the order $R$ by a simple $\le$ sign. Then $a\le b$ and $c\le d$ are equivalent to $a-b\le 0$ and $0\le d-c$, hence $a-b\le d-c$ or $a +c \le b+d$.

Nothing special is going on here when comparing to the standard ordering on the reals.