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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