properties of products for flatness in the category of $s$-posets

this paper is devoted to the study of products of classes of right $s$-posets possessing one of the flatness properties and preservation of such properties under products. specifically, we characterize a pomonoid $s$ over which its nonempty products as right $s$-posets satisfy some known flatness properties. generalizing this results, we investigate products of right $s$-posets satisfying condition $(pwp)$. finally, we investigate pomonoids over which products of right $s$-posets transfer an arbitrary flatness property, projectivity, freeness, and regularity to their components.