Evaluating characterizations of homomorphisms on truncated vector lattices of functions

Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $\psi$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% $$\psi (f\wedge {\mathbf{1}}_X)=\min \{\psi \left(f\right),1\}\text{ for all }f\in L\mathrm{.}$$ We prove that a linear functional $\psi$ on $L$ is a truncation homomorphism if and only if $\psi$ is a lattice homomorphism and% $$\sup \{\psi \left(f\right):f\le {\mathbf{1}}_X\}=1.$$ This allows us to prove different evaluating characterizations of truncation homomorphisms. In this regard, a special attention is paid to the continuous case and various results from the existing literature are generalized.