A classification of peak-pit maximal Condorcet domains.

In this paper, we introduce a weaker notion of separability for set-systems and demonstrate that the class of maximal weakly separated systems precisely corresponds to the class of peak-pit maximal Condorcet domains. Additionally, we present a generalisation of arrangements of pseudolines and establish that the sets of chamber sets from them coincide with maximal weakly separated systems, enabling the construction of all peak-pit maximal Condorcet domains. Furthermore, we reveal that peak-pit maximal Condorcet domains coincide with connected maximal Condorcet domains. • Peak-pit maximal Condorcet domains are characterised by weakly separated ideals. • Weakly separated ideals are characterised by generalised arrangements of pseudolines. • Peak-pit maximal Condorcet domains coincide with connected maximal Condorcet domains. • Peak-pit maximal Condorcet domains are copious and ample. • The size of the ideal of any peak-pit maximal Condorcet domain on [n] is n + 1 2 + 1. [ABSTRACT FROM AUTHOR]