Condorcet domains and distributive lattices

Condorcet domains are sets of linear orders where Condorcet's effect can never occur. Works of Abello, Chameni-Nembua, Fishburn and Galambos and Reiner have allowed a strong understanding of a significant class of Condorcet domains which are distributive lattices -in fact covering distributive sublattices of the permutoèdre lattice- and which can be obtained from a maximal chain of this lattice. We describe this class and we study three particular types of such Condorcet domains.