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.

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.