# Adjective describing a property inherited by factors

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 $\pi\circ f = g\circ \pi$.

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(1) There are hereditary rings. You can't count on a word to have a fixed meaning across fields of mathematics. (2) "Factor" in what sense? (3) As the bumper sticker says: Insanity is hereditary; you get it from your kids. – Tom Goodwillie Oct 2 '10 at 4:06
@Tom: (1) Fair point. The first paragraph is purely by way of motivation -- the second paragraph is the real content of the question. My interest is in dynamical systems, but I'd be happy to know what terminology may be used anywhere. (2) For me, "factor" in the sense of a semi-conjugacy of dynamical systems: that is, given two compact metric spaces $X$ and $Y$ and two continuous maps $f\colon X\to X$ and $g\colon Y\to Y$, a surjective map $\pi\colon X\to Y$ such that $\pi \circ f = g\circ \pi$. But factor in any sense would be a start. (3) Sadly, I have no witty reply... – Vaughn Climenhaga Oct 2 '10 at 4:26
(2) Sorry, I still don't even know whether you are calling $\pi$ a factor of something or $(X,f)$ a factor of something or what. Anyway, to me there would be nothing wrong with saying "Let us call a property $P$ hereditary if whenever a [blank1] $X$ has $P$ and $Y$ is a [blank2] of $X$ then $Y$ has $P$." Just as you might introduce the term "regular" or "good" or "of the second kind" in a particular limited context. – Tom Goodwillie Oct 2 '10 at 4:44
(3) I wasn't being successfully witty myself. But in the back of my mind was the thought that "factor" might possibly be some sort of antithesis of "subobject" for you, in which you would be in the market for some word for a backwards kind of inheriting process. – Tom Goodwillie Oct 2 '10 at 4:44
@Climenhaga: If you end up having to invent some new terminology, I would go for something descriptive, like "quotient-hereditary" or "factor-hereditary", and avoid words like "cohereditary" or "contahereditary". – André Henriques Oct 2 '10 at 20:00