On additivity of mappings on measurable functions

We prove the additivity of regular $l$-additive mappings $T\colon\mathscr K\to[0,+\infty]$ of a hereditary cone $\mathscr K$ in the space of measurable functions on a measure space. Some examples are constructed of non-$d$-additive $l$-additive mappings $T$. The monotonicity of $l$-additive mappings $T\colon\mathscr K\to[0,+\infty]$ is established. The examples are constructed of nonmonotone $d$-additive mappings $T$.
Let $(S,+)$ be a commutative cancellation semigroup. Given a mapping $T\colon\mathscr K\to S$, we prove the equivalence of additivity and $l$-additivity. It is shown that a strongly regular $2$-homogeneous $l$-subadditive mapping $T$ is subadditive. All results are new even in case $\mathscr K=L^+_\infty$.