Definition:Subadditive Function (Conventional)

Definition
Let $$\left({S, +_S}\right)$$ and $$\left({T, +_T; \preceq}\right)$$ be semigroups such that $$\left({T, +_T; \preceq}\right)$$ is ordered.

Let $$f: S \to T$$ be a mapping from $$S$$ to $$T$$ which satisfies the relation:
 * $$\forall a, b \in S: f \left({a +_S b}\right) \preceq f \left({a}\right) +_T f \left({b}\right)$$

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.