"Down-Closed", "Down Ideal", Something Else?

Question

Let $X$ be an a set and let $\mathcal{C}$ be a collection of subsets of $X$ satisfying the following property:

If $A$ and $A^\prime$ are subsets of $X$ with $A \in \mathcal{C}$ and $A^\prime \subseteq A$, then $A^\prime \in \mathcal{C}$.

I have heard this described variously as "$\mathcal{A}$ is down-closed" and "$\mathcal{A}$ is a down ideal", but neither of these phrases seem very prevalent on the internet. Is there a more common name for this property?

Answer 1

As this page indicates, you're missing a condition required for the given collection to be an 'ideal'. As this page indicates, the terms 'downward closed', 'down set', 'lower set', et al. are appropriate in this case.

Answer 2

The term "downset" is sometimes used. See e.g. Anderson, "Combinatorics of Finite Sets".

Answer 3

In "The Probabilistic Method", Alon and Spencer call such a collection "monotone decreasing".

Answer 4

In the case where $\mathcal{C}$ consists of finite sets only, $\mathcal{C}$ is an abstract simplicial complex.