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

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, thereby providing new combinatorial and axiomatic characterizations of these domains. We further show that regular vines are equivalent to $(n,3)$-extremal lattices from formal concept analysis, and that these lattices are in turn equivalent to extremal binary matrices with no triangles from combinatorial matrix theory. Consequently, these lattices and matrices also fit naturally into the same splitting-and-merging framework, providing further examples from different areas unified by our axiomatic characterization.