Embeddings of metric Boolean algebras in ℝ^N

2022/07/29 by Bonzio, Stefano, Loi, Andrea
#FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)

paper · doi:10.48550/arxiv.2207.14600

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¬ b)\vee(¬ a\wedge b)), for any a,b∈ A, sometimes referred to as 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 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 (At(\A), d_m) embeds isometrically in ℝ^N.

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