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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

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  • $\begingroup$ What does it mean for a subposet to be a forest? $\endgroup$
    – Sean Tilson
    Commented Aug 31, 2016 at 9:03
  • $\begingroup$ Every element is covered by at most one element. $\endgroup$
    – Richard
    Commented Aug 31, 2016 at 9:04

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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.

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    $\begingroup$ Any finite distributive lattice has Möbius numbers in $\{-1,0,1\}$. $\endgroup$
    – Richard Stanley
    Commented Sep 6, 2016 at 14:49

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