In my research, I stumbled upon a particular kind of poset and I was wondering, whether there is something in the literature (I could not find anything so far). 


They are distributive lattice $L$ such that their associated subposet of join-irreducibles is a forest (so $L$ is the Birkhoff lattice of a forest). Is there anything known in the literature about these lattices?


Thanks!


Richard



Birkhoff Lattice of a forest
This set of lattices is the closure of the set {poset with one element} under two allowed operations: adding a top element or taking a Cartesian product. This implies that the Möbius numbers are in $\{-1,0,1\}$. Any similar property whose behavior under the two operations can be easily followed can also be described on the familly.


EDIT: I have met them at least twice in my own research, first in a work on operads (<https://hal-univ-paris13.archives-ouvertes.fr/hal-00165245/document>) and more recently in <https://hal.archives-ouvertes.fr/hal-01339996>.


