What's the name of this type of a set?

Question

So I have a set $\{i_1,i_3,i_5\}$. What do we call the following set? Is there a standard name for it? $\emptyset, \{i_1\}, \{i_1,i_3\}, \{i_1,i_3,i_5\}$. Note that we do not have $\{i_3,i_5\}$ in it so it is not a power set. It seems like it is a "naturally ordered poset" to me.

Answer 1

It seems like you're really starting with an ordered set $(S,<)$. That is, if you just started with an arbitrary set with three elements $\{x,5,*\}$, there would be nothing to distinguish $\{x,*\}$ from $\{x,5\}$. It's important that you know $i_1$ comes first, followed by $i_3$, then $i_5$, right?

Then what you've written down is the set of all downwards-closed subsets of $S$. In general, given $(S,<)$, you can form $\{X\in\mathcal{P}(S)\mid \text{if }a\in X\text{ and }b<a\text{ then }b\in X\}$.

I'm extrapolating here, since you haven't made your intentions really clear. Does this capture what you had in mind?