In my thesis I encountered the following problem: suppose that $E,F$ are Riesz spaces such that $E$ is a subspace of $F$ and that the ordering on $E$ matches the one on $F$, i.e. $x\leq_E y\Rightarrow x\leq_F y$ where $\leq_E$ is the ordering on $E$ and $\leq_F$ the one on $F$. My question is: for any element $x\in E$ we can define the positive part in $E$ and we can define it in $F$. Are those two elements necessarily the same?
If this is true, can we drop the condition that $E$ is Riesz and just assume that we have an element $x\in E$ for which the positive part in $E$ exists?
In most simple examples I tried I found the answer to be yes, but I could not find a proof. Has anyone here encountered a similar problem and found a solution? Thanks in advance.
This result is false. Indeed, let $E$ be the Riesz space $\mathbb{R}^{3}$ provided with the pointwise algebraic and lattice operations and $F=\left\{ \left( x,y,x+y\right) \text{ }/\text{ }x,y\in \mathbb{R}\right\} .$ Then $F$ , equipped with the algebraic and lattice operations inherited from $E,$ is a Riesz space. However the positive part of $\left( x,y,x+y\right) $ in $E$ equals $\left( x^{+},y^{+},\left( x+y\right) ^{+}\right) $ and the positive part of $\left( x,y,x+y\right) $ in $F$ equals $\left( x^{+},y^{+},x^{+}+y^{+}\right) .$