TITLE: Terminology question on sets and intersections
QUESTION [5 upvotes]: Is there a terminology for the following property:
Given $n$ sets, that for all combinations of two of them, $A_i$, $A_j$, the intersection $A_i \cap A_j$ is either the null set or equal to either $A_i$ or $A_j$. 
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The picture I had in my head was lego blocks representing the different sets. 
Set-up A : Picture link doesn't satisfy (Red, Yellow is just one example)
Set-up B  Picture below satisfies the property. Intersection of any two blocks is either empty or one of the two blocks in its entirety.

REPLY [2 votes]: Let $\mathscr{A}$ be the family of sets. Your conditions amounts to saying that the partial order $\langle\mathscr{A},\supseteq\rangle$ is a directed forest. The example in your picture is even an arborescence (directed tree):
                                * light brown  
                               / \  
                              /   \  
                      yellow *     * dark brown  
                                  / \  
                                 /   \  
                            red *     * blue