Generalisation of the Danilov-Karzanov-Koshevoy Construction for Peak-Pit Condorcet Domains

TL;DR

This paper extends the algebraic understanding of the composition operation on tiling Condorcet domains, providing formulas for their size and challenging previous assumptions about maximal peak-pit Condorcet domains.

Contribution

It offers an algebraic formulation of the composition operation and refines the understanding of the maximal size of peak-pit Condorcet domains, disproving earlier conjectures.

Findings

Provided a formula for the size of composed Condorcet domains

Disproved Fishburn's conjecture on maximal peak-pit Condorcet domains

Enhanced the algebraic framework for Condorcet domain analysis

Abstract

Danilov, Karzanov and Koshevoy (2012) geometrically introduced an interesting operation of composition on tiling Condorcet domains and using it they disproved a long-standing problem of Fishburn about the maximal size of connected Condorcet domains. We give an algebraic definition of this operation and investigate its properties. We give a precise formula for the cardinality of composition of two Condorcet domains and improve the Danilov, Karzanov and Koshevoy result showing that Fishburn's alternating scheme does not always define a largest peak-pit Condorcet domain.