Definition:Subadditive Function (Conventional)

Definition

Let $\struct {S, +_S}$ and $\struct {T, +_T, \preceq}$ be semigroups such that $\struct {T, +_T, \preceq}$ is ordered.

Let $f: S \to T$ be a mapping from $S$ to $T$ which satisfies the relation:

$\forall a, b \in S: \map f {a +_S b} \preceq \map f a +_T \map f b$

Then $f$ is defined as being subadditive.

The usual context in which this is encountered is where $S$ and $T$ are both the set of real numbers $\R$ (or a subset of them).

Also see

Compare with the field of measure theory, in which the definition of subadditive function is completely different.

• Results about subadditive functions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): additive function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): additive function