TITLE: Terminology for the property $f(2x) = 2f(x)$
QUESTION [5 upvotes]: The property $$f(2x) = 2f(x)$$ is strictly weaker than linearity: linearity implies it, but if e.g. $f : \mathbb{N} \to \mathbb{N}$ then $f$ can satisfy this property without being linear, as exemplified by OEIS A006519: $x \to 2^{\nu_2(x)}$.
Does this property have a name?

REPLY [2 votes]: For a given $\alpha$, let $M_\alpha$ be the commutative monoid formed by the powers of $\alpha$ under multiplication.
Any set $X$ such that if $x \in X$ then $\alpha x \in X$ can be seen as an $M_\alpha$-set in which the action is multiplication.
Then, a function $f \colon X \to X$ such that $f(\alpha x) = \alpha f(x)$ is simply an $M_\alpha$-endomorphism.