Algebraic form of aЁtheorem of Hahn–Banach type for lattice manifolds

Let $E$ be a vector lattice. A linear functionalf on $E$ is called a lattice homomorphism if $f(\sup(x,y))=\max(f(x),f(y))$ for all $x,y\in E$. For lattice homomorphisms a theorem of Hahn–Banach type is valid. In this note we prove an algebraic analog of this theorem.