Abstract
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving −1 and/or +1, so-called residuations. New definitions and results for directional derivatives, subdifferentials and Legendre–Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for set-valued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by non-negative real numbers. An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving -infinity and/or +infinity, so-called residuations. Based on this, definitions and results for directional derivatives, subdifferentials and Legendre--Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for set-valued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by non-negative real numbers.