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Let $P$ be a finite poset (partially ordered set).

I am wondering whether the following condition on $P$ has been studied somewhere:

(#) No interval $[a,b]$ in $P$ has $3$ elements.

Note that intervals with $3$ elements are necessarily isomorphic to a total order on $3$ elements.

For example, the condition (#) holds in the standard posets of set partitions and noncrossing partitions. It does not hold in the Tamari lattices.

Condition (#) implies that no interval in $P$ can be isomorphic to a total order of cardinality of least $3$. So intervals of cardinality at least $3$ in these posets have an antichain of size at least $2$.

There is a classical notion of thin poset, where intervals of length 2 are isomorphic to a boolean lattice of cardinality $4$. This implies (#).

So my question is

Is there an established name for the condition (#) ? Did it appear somewhere ?

This should not be confused with the avoidance of $3$-chains $a < b < c$, which is much more restrictive.

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    $\begingroup$ Relevant discussion: mathoverflow.net/questions/330335 $\endgroup$
    – Gjergji Zaimi
    Commented May 16, 2021 at 19:40
  • $\begingroup$ Thanks. So this is known as 2-thick posets. And Eulerian posets are 2-thick. $\endgroup$
    – F. C.
    Commented May 17, 2021 at 6:38
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    $\begingroup$ See Theorem 7 of Logan and Shahriari, "A new matching property for posets and existence of disjoint chains." $\endgroup$
    – Tri
    Commented Sep 5, 2021 at 20:35
  • $\begingroup$ Logan and Shahriari is here: doi.org/10.1016/j.jcta.2004.06.002 $\endgroup$
    – F. C.
    Commented Oct 1, 2021 at 17:03
  • $\begingroup$ See Theorem 2 of Bjorner, "On complements in lattices of finite length," Discrete Mathematics 36 (1981), 325-326. $\endgroup$
    – Tri
    Commented Mar 13, 2022 at 19:04

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