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(I didn't see any question like this on the CSTheory exchange -- apologies if it has already been answered.)

In a nested hierarchy, a higher level includes all levels below it.

The classic example of a "non-nested" hierarchy is military rank. A general is not a lieutenant, so military rank is a non-nested hierarchy.

But here's my objection -- doesn't a general have all the skills a lieutenant has, plus more skills? So in that sense, isn't every general a lieutenant? He may not be called "lieutenant", but in reality he is a lieutenant-plus.

Thus, I am wondering... can we really say that any hierarchy is non-nested?

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    $\begingroup$ This seems off-topic unless you add a connection to research-level theoretical computer science. $\endgroup$
    – Laakeri
    Dec 22 '20 at 18:56
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    $\begingroup$ I thought from the title it was going to ask about hierarchies like PH or the W-hierarchy, and whether there are analogous hierarchies that aren't linearly ordered... $\endgroup$ Dec 22 '20 at 21:51
  • $\begingroup$ Sorry guys - I’ll try to move this question to the StackExchange for the English language (or for Math). I’m a data wrangler rather than a true CS person. $\endgroup$
    – Steve_P
    Jan 6 at 19:27
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I think this question is more about the semantics of the english language than about cs theory. Unless this has a connection to some problem. I think generally speaking, hierarchy of things just means there is a well ordered relation of these things. And nested just means that the things are nested within each other. The relation is defined based on nesting. So the military ranks are a hierarchy, but not a nested one because the groups aren't nested within each other.

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  • $\begingroup$ Thanks @styopa ! Can I ask a quick follow-up… A, B, C, and D are sets. A includes B. B includes C and D. Thus, for nesting-level: A = 0. B = 1. C = 2. D = 2. Nesting-level is a total order on A, but it is not a well-order because not every subset of A has a least element. A is nested, but can we call it a hierarchy? $\endgroup$
    – Steve_P
    Jan 6 at 19:28
  • $\begingroup$ From the definition of hierarchy I've seen most, I wouldn't call it a hierarchy. However, I don't think this matters for any practical purposes. You could just make a small note before calling it a hierarchy. $\endgroup$
    – styopa
    Jan 8 at 0:08
  • $\begingroup$ Thank you so much! @styopa $\endgroup$
    – Steve_P
    Jan 9 at 6:50

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