Gelfand-Phillips-type properties in Banach lattices

We study p-limited and almost p-limited sets in Banach lattices and their connections with relatively p-compact and relatively compact sets. We investigate the weak and the strong Gelfand-Phillips property of order p, as well as the p-GP property introduced by Delgado and Piñeiro, providing conditions under which these properties may coincide. Additionally, we prove that a Banach lattice E is a KB-space if and only if every almost p-limited set in E is relatively weakly compact if and only if every the adjoint of a weakly compact taking values on E is disjoint p-summing.