On the FPV Property, Monotone Operator Structure and the Monotone Polar of Representable Monotone Sets

We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related point. This study allows us to prove that all bigger conjugate representable monotone sets are monotonically closed. This result sheds light on the structure of the domains for maximal monotone operators and enables us to study the sum theorem for FPV operators in Banach spaces which posses a dual space that has a strictly convex renorm.