A note on the representation of lattice homomorphisms

The purpose of this note is to specify for lattice homomorphisms Riesz type representation theorems obtained by Wright and Coquand for positive operators. We prove that every lattice homomorphism acting from a vector lattice of continuous functions on a Hausdorff compact topological space with values in a Dedekind complete or Dedekind \(\sigma \) $\sigma$ -complete vector lattice admits an integral representation by a quasi-regular vector measure that is a \(\sigma \) $\sigma$ -continuous Boolean homomorphism. We use an operator approach to obtain suitable vector measures to represent lattice homomorphisms.