Linear functionals preserving the units of a Riesz space

Let $L$ be a Riesz space with a strong unit $e>0$.$\mathfrak{\ }$We show that a unital linear functional $H:L\rightarrow \mathbb{R}$ satisfies $$ for any strong unit $u\in L$ if and only if $H$ acts like a Riesz homomorphism on every $e$-clean vector subspace of $L.$ We deduce that $L$ is $e$-clean if and only if any unital linear functional $$ such that $H\left( u\right) \neq 0$ for any strong unit $u\in L$ is a Riesz homomorphism.