Embeddings of metric Boolean algebras in $\mathbb{R}^{N}$

TL;DR

This paper investigates the conditions under which the atom space of a finite metric Boolean algebra can be isometrically embedded into Euclidean space, linking measure topology with geometric embedding properties.

Contribution

It characterizes the measures on finite Boolean algebras that allow their atom spaces to embed isometrically into Euclidean space.

Findings

Characterization of measure topologies enabling Euclidean embeddings

Conditions for isometric embedding of atom spaces in finite Boolean algebras

Insights into metric properties of Boolean algebra atom spaces

Abstract

A Boolean algebra $\A$ equipped with a (finitely-additive) positive probability measure $m$ can be turned into a metric space $(\A , d_{m})$, where $d_{m}(a,b)= m ((a\wedge\neg b)\vee(\neg a\wedge b))$, for any $a,b\in A$, sometimes referred to as \emph{metric Boolean algebra}. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in $\mathbb{R}^{N}$ (for a certain $N$) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra $\A$ such that the metric space $(\mathsf{At}(\A), d_m)$ embeds isometrically in $\mathbb{R}^{N}$.