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Does this property have a name?

Given a collection of sets $\mathcal{P}$, for all pairs $A, B\in\mathcal{P}$, either $A\cap B=\emptyset$ or $A\subseteq B$ or $B\subseteq A$.

This concept could equally apply to monoids, groups, partial orders or other mathematical structures, with some adjustments to the definitions. For instance, for monoids and groups we would replace $A\cap B=\emptyset$ by $A\cap B=\{\epsilon\}$, where $\epsilon$ is the unit of the monoid/group.

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    $\begingroup$ Embarassing: I asked the same question before. $\endgroup$ – Dave Clarke Sep 12 '13 at 10:10
  • $\begingroup$ In the light of our recent discussion, I think I have to ask: what is the relation to CS here, that is which concept are you trying to model? $\endgroup$ – Raphael Sep 16 '13 at 7:24
  • $\begingroup$ @Raphael: The concept came up while investigating models of software product lines. I distilled the essence, and chose more basic mathematical structures as the basis of the question. Had I included all the details about software product lines in order to phrase the question, ... well, it wouldn't have been so easy to get a clear answer so quickly. $\endgroup$ – Dave Clarke Sep 16 '13 at 10:53
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I think that's called a laminar family.

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    $\begingroup$ mathoverflow.net/a/26696/10833 $\endgroup$ – scaaahu Sep 12 '13 at 9:57
  • $\begingroup$ @DaveClarke I think we should thank adrianN. I have been looking for the name of that property myself for a while. I missed your previous question and I never bother to ask. This property is hard to search. Thanks to adrianN again. $\endgroup$ – scaaahu Sep 12 '13 at 10:52
  • $\begingroup$ Everybody seems to be happy, but can you please give a reference so the answer holds more weight for future visitors? (The link to MO does not quite count.) $\endgroup$ – Raphael Sep 16 '13 at 7:22

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