Abstract convex theory for set-valued sub-topical mappings

Sub-topical functions constitute a significant class of abstract convex functions and play a pivotal role in optimization theory. However, the scalar framework of sub-topical functions faces limitations when addressing set-valued optimization problems equipped with order structures. To bridge this gap, this paper generalizes the concept of sub-topicality from scalar-valued mappings to set-valued mappings. By leveraging the notion of weak supremum in vector spaces, we first construct a class of set-valued elementary mappings with specific algebraic properties to serve as supporting elements. Subsequently, we introduce the support set for these set-valued sub-topical mappings and establish their upper envelope representation with respect to this family of elementary mappings. Finally, we develop a comprehensive abstract convexity theory for set-valued sub-topical mappings. This framework lays a solid theoretical foundation for future research in set-valued optimization and related fields.