A New Cardinal Invariant Related to Adding Real Functions

Abstract

Let $F\subseteq\mathbb{R}^{\mathbb{R}}$. The additivity of $F$, briefly A$(F)$, is the minimum cardinality of a family $G\subseteq\mathbb{R}^{\mathbb{R}}$ with the property that $h+G\subseteq F$ for no $h\in\mathbb{R}^{\mathbb{R}}$. In this paper we consider the notion of super-additivity which we will denote by $A^*$. If $F\subseteq\mathbb{R}^{\mathbb{R}}$, then $A^*(F)$ is the minimum cardinality of a family of functions $G$ with the property that for any $H\subseteq\mathbb{R}^{\mathbb{R}}$ if $|H|<A(F)$, there is a $g\in G$ such that $g+H\subseteq F$. We calculate the super-additivities of the families of Darboux-like functions and their complements.