Evaluating characterizations of truncation homomorphisms on truncated vector lattices of functions

Abstract

Let X be a nonempty set. A vector sublattice L of $\mathbb {R}^{X}$ is said to be truncated if L contains with any function f the function $f\wedge \mathbf {1}_{X}$. A nonzero linear functional $\psi$ on L is called a truncation homomorphism if it preserves truncation (i.e., $\psi \left( f\wedge \mathbf {1}_{X}\right) =\min \left\{ \psi \left( f\right) ,1\right\}$ for all $f\in L$). These concepts generalize the notion of unital lattice homomorphisms on unital vector sublattices of $\mathbb {R}^{X}$ . Via a unitization process, we extend the different evaluating characterizations of unital lattice homomorphisms, previously obtained by Garrido and Jaramillo, to the truncation homomorphisms on truncated vector sublattices of functions.