Arrow type impossibility theorems over median algebras

TL;DR

This paper characterizes trees as median algebras and explores Arrow type impossibility theorems for median algebra mappings, revealing that such theorems hold only when the target algebra is a tree.

Contribution

It introduces a characterization of trees within median algebras and establishes conditions under which Arrow type impossibility theorems apply to median algebra mappings.

Findings

Trees are characterized as median algebras and semilattices.

Arrow type impossibility theorems hold only when the target median algebra is a tree.

Median homomorphisms between product median algebras are described.

Abstract

We characterize trees as median algebras and semilattices by relaxing conservativeness. Moreover, we describe median homomorphisms between products of median algebras and show that Arrow type impossibility theorems for mappings from a product $\mathbf{A}_1\times \cdots \times \mathbf{A}_n$ of median algebras to a median algebra $\mathbf{B}$ are possible if and only if $\mathbf{B}$ is a tree, when thought of as an ordered structure.