Orthogonally Bi-additive Operators-II

In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map T : I -> W defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space E x F. Then we prove that a positive orthogonally biadditive operator T : E x F -> W is laterally-to-order continuous if and only if the kernel of each S with 0 <= S <= T is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator T : E x F -> W.