  We study the class of Banach lattices that are positively polynomially Schur.
Plenty of examples and counterexamples are provided, lattice properties of this
class are proved, arbitrary $L_p(\mu)$-spaces are shown to be positively
polynomially Schur, lattice analogues of results on Banach spaces are obtained
and relationships with the positive Schur and the weak Dunford-Pettis
properties are established.
