In a text I am using it states the following: "The set of all Riesz homomorphisms between two Riesz spaces does not ordinarily have a simple structure of its own. Consider for example, the set of Riesz homomorphisms from $\mathbb{R}^{2}$ to $\mathbb{R}$, which is not closed under ordinary addition."
Can anyone see how it is clear that the set of Riesz homomorphisms from $\mathbb{R}^{2}$ to $\mathbb{R}$ is not closed under ordinary addition? Is there a clear example which shows that this set is not closed under addition?
Thanks for any assistance.
As the supremum in $\mathbb R^2$ is taken pointwise, i.e., $(x,y)\vee (x',y')=(x\vee x',y\vee y')$, the projections $p_1$ and $p_2$ onto the first and second variable are Riesz homomorphisms.
However, their sum is not: You can easily find $(x,y),(x',y')\in\mathbb R^2$ such that $(x+y)\vee (x'+y')\neq (x\vee x')+(y\vee y')$.