An axiomatic framework from splitting and merging in MAT-labeled graphs, vines, and single-peaked domains

TL;DR

This paper develops an axiomatic framework based on splitting and merging operations to unify the combinatorial structures of MAT-labeled graphs, vines, and single-peaked domains, resolving an open problem in economics.

Contribution

It introduces a novel axiomatic characterization of these structures using combinatorial species, linking concepts across social choice theory, probability, and formal concept analysis.

Findings

Established explicit correspondences between single-peaked domains, MAT-labeled graphs, and vines.

Provided an axiomatic framework that uniquely characterizes these structures.

Connected regular vines to extremal lattices from formal concept analysis.

Abstract

In recent work (Forum Math.~Sigma, 2024), we established a correspondence between MAT-labeled graphs arising from hyperplane arrangements and regular vines from probability theory. In this paper, we extend this connection to Arrow's single-peaked domains in social choice theory. We show that MAT-labeled complete graphs, regular vines, and maximal Arrow's single-peaked domains arise from the same recursive combinatorial structure. Our main result gives an axiomatic characterization of these objects using the language of combinatorial species. At the heart of this characterization are two fundamental operations, called splitting and merging, together with natural compatibility conditions that uniquely determine the structures. As consequences, we obtain explicit correspondences between maximal Arrow's single-peaked domains, MAT-labeled complete graphs, and regular vines, resolving an open problem in the economics literature concerning the combinatorial characterization of single-peaked domains. We further show, by a direct proof, that regular vines are equivalent to $(n,3)$-extremal lattices from formal concept analysis. Consequently, these extremal lattices also fit naturally into the same splitting and merging framework, providing another example from a different area that satisfies our axiomatic characterization.