Given some class of objects, a property P is called "hereditary" if it is such that whenever X has P and Y is a subobject of X, then Y has P as well. (At least, this is my understanding of the meaning of "hereditary" -- please correct me if I'm wrong.)
Is there a word in common usage to describe a property that is passed to factors? That is, what word should we use to describe a property P such that whenever X has P and Y is a factor of X, then Y has P as well?
Edit: By "Y is a factor of X" I mean that there is a surjective map $\pi\colon X\to Y$ that preserves the structure of the objects $X$ and $Y$ (whether the structure is that of a group, a topological space, a dynamical system, or whatever else you may be interested in). As pointed out in the comments, this may also be called a quotient in some contexts. In the setting I'm most interested in, $X$ and $Y$ are topological dynamical systems -- compact metric spaces with continuous maps $f\colon X\to X$ and $g\colon Y\to Y$ -- and $\pi$ preserves the dynamics in the sense that $\pi\circ f = g\circ \pi$.