# About finite posets without intervals of size 3

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.