The lateral order on Riesz spaces and orthogonally additive operators. II

The present paper aims to describe the relationships between the intersection property, introduced and studied in the previous paper by the authors, with other known properties of Riesz spaces, and to prove that every lateral ideal of a Riesz space is a kernel of some positive orthogonally additive operator (it is easy to see that the kernel of every positive orthogonally additive operator is a lateral ideal). We provide examples of Riesz spaces with the principal projection property (and hence, with the intersection property) which fail to be C-complete. The above results give complete answers to problems posed in the first part of the present paper by the authors.