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I can think of two nonequivalent ways of defining an interval in a poset:

  1. An interval of a poset $P$ is a subset $I\subset P$ with the property that for all $x, y, z\in P$ such that $x < y < z$ and $x, z\in I$, we have that $y\in I$.

  2. An interval of a poset $P$ is a subset $I\subset P$ of one of the 9 forms: $P$, $\{\,y\in P\mid x < y\,\}$, $\{\,y\in P\mid x\le y\,\}$, $\{\,y\in P\mid y < z\,\}$, $\{\,y\in P\mid y\le z\,\}$, $\{\,y\in P\mid x < y < z\,\}$, $\{\,y\in P\mid x\le y < z\,\}$, $\{\,y\in P\mid x < y\le z\,\}$, $\{\,y\in P\mid x\le y\le z\,\}$, where $x, z\in P$.

Is there a consensus about which of these two is the "right" one? Respectable references are welcome.

I would think that the first definition is better, but it is the second that is given in Bourbaki's Theory of Sets and on nLab wiki.

An additional related question: does an interval have to be nonempty?

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  • $\begingroup$ The usual definition is the first one. $\endgroup$
    – jjagmath
    Commented Dec 11, 2022 at 12:23
  • $\begingroup$ @jjagmath, could you provide some reference, please? I saw the second one two: ncatlab.org/nlab/show/interval. Also, does it have to be nonempty? $\endgroup$
    – Alexey
    Commented Dec 11, 2022 at 12:26
  • $\begingroup$ An interval in the first sense is not always defined by "tight bounds" in the second sense. Consider the set of rational numbers whose square is smaller than 2. $\endgroup$
    – Karl
    Commented Dec 11, 2022 at 15:19
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    $\begingroup$ I understand both definitions are used by different authors, but I prefer the second one, while the first is the definition of convex subset. I give some references to justify this preference in my answer to the question What is an interval of a lattice? (Those references seem to apply equally to lattices or posets in general.) $\endgroup$
    – amrsa
    Commented Dec 11, 2022 at 15:24
  • $\begingroup$ @Karl, that's what I mean by "nonequivalent." $\endgroup$
    – Alexey
    Commented Dec 11, 2022 at 16:12

1 Answer 1

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The first definition is fine if you are dealing with a total order, but it doesn't match my intuition for what an interval is otherwise. For example, let us take $\mathcal{P}(\mathbb{N})$ partially ordered by $\subset$.

I would not consider $\{\{2\},\{3\},\{4\}\}$ to be an interval here. But it matches Definition 1. Something like $\{\{2\},\{2,3\},\{2,679\},\{2,3,679\}\}$ makes for a much better interval.

Whether the empty set should be considered an interval or not really varies from setting to setting.

So, for interval, I'd go with Definition 2. Definition 1 describes a convex set instead.

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  • $\begingroup$ Well, in $\Bbb Q$ the set $\{q\mid 2<q^2<3\}$ is covex, but it is not an interval since it lacks endpoints. $\endgroup$
    – Asaf Karagila
    Commented Dec 12, 2022 at 0:52
  • $\begingroup$ @AsafKaragila True and annoying. But I'd rather have that not be an interval than have any antichain be one. If need be, one could of course move to some completion and take the intervals there. $\endgroup$
    – Arno
    Commented Dec 12, 2022 at 0:54
  • $\begingroup$ Oh I'm not disagreeing with that. $\endgroup$
    – Asaf Karagila
    Commented Dec 12, 2022 at 8:06

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