# Name for a function whose effect is canceled by another function?

I've been saying function $$f$$ "eclipses" $$g$$ if $$f(g(x)) = f(x)$$ for all $$x$$. For example, if $$f(x) = \lvert x \rvert$$ and $$g(x) = -x$$, then $$f$$ eclipses $$g$$.

Is there an established word for this property?

• Don't know that there is one. "Eclipse" is as good a candidate for a name as any.... – fleablood Jun 16 at 16:47
• This is loosely related to the concept of en.wikipedia.org/wiki/Coequalizer, so maybe one could call $f$ "co-equalizing $g$ and the identity" – Hagen von Eitzen Jun 16 at 17:08

I would express this as "$$f$$ is invariant under $$g$$" or "$$f$$ is $$g$$-invariant", because of the analogy with group actions:
Compare this with: If $$G$$ is a group acting linearly on a vector space $$V$$, it induces an action on the dual $$V^*$$ by letting $$g*f = f \circ g$$. If $$g*f = f$$ for all $$g \in G$$, we would say $$f$$ is $$G$$-invariant.
Now, if $$f : X \to Y$$ and $$g : X \to X$$, then $$g$$ determines a monoid action of $$\mathbb N$$ on $$X$$ by letting $$n$$ act by $$g_n = g \circ \cdots \circ g$$. The monoid action induces a monoid action on the set of functions $$X \to Y$$, by letting $$n*f = f \circ g_n$$. A function $$f$$ is invariant under this monoid action iff $$f \circ g = f$$.