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From Wikipedia (Note that below I corrected a place which I think is a typo):

A set system $(E, F)$ is a collection $F$ of subsets of a ground set $E$. $(E, F)$ is said to have the Interval Property, if $A, B, C ∈ F$ with $A ⊆ B ⊆ C$, then, for all $x ∈ E \setminus C$, $A\cup \{x\} ∈ F$ and $C\cup \{x\} ∈ F$ implies $B\cup\{x\} ∈ F$.

Does the name "the Interval Property" suggest it comes from some properties of intervals on $\mathbb{R}$? Otherwise, how well can the interval property be understood?

Thanks and regards!

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This is my guess. The interval property generalizes a certain behavior of intervals in $\mathbb{R}$. If $A$ is something like an interval and $A \cup \{ x \} $ is also something like an interval then either $x \in A$ or $x$ is something like an endpoint for $A$. For example $(0, 1) \cup \{ 1 \} = (0, 1]$, another interval.

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Thanks! That makes sense. What structures does the collection of intervals of $\mathbb{R}$ have? –  Tim Jan 3 '13 at 22:21
    
Locally it seems to be diamonds. Suppose that $x < y$. Using inclusion as the partial order any interval between $(x, y)$ and $[x, y]$ is either $[x, y)$ or $(x, y]$. This seems to suggest that we can just look at, say, closed intervals and then add in the other type of intervals. Closed intervals seem to be ordered like pair of numbers where the first number is no larger than the second number. Notice that in my example if $x = y$ then three of the intervals are empty. A certain amount of additional work is required. –  Jay Jan 4 '13 at 1:30
    
Thanks! What is the definition of diamonds as a math structure? I didn't find Wikipedia has it. –  Tim Jan 4 '13 at 1:36
    
Diamond's is a made up name. Thing on the top, thing on the bottom, two things in between. It looks like a diamond. –  Jay Jan 4 '13 at 23:17
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