Vector conjugation and subdifferential of vector topical function in complete lattice

In our previous work, we introduced a vector topical function which takes values in a general ordered Banach space. With some weak versions of supremum, a set-valued mapping was proposed to build the envelope for it. In this paper, regarding the image space as a complete lattice, we present a vector mapping as the support to study the abstract convex framework of the vector topical function. This framework provides a dual space consisting of vector mappings. Based on that, we obtain the global Lipschitz of the vector topical mapping, discussing the abstract convex conjugation and subdifferential. Further more, we investigate how the vector topical feature behaves from the dual point of view. Some dual characterizations of vector topical mapping are also established.