The complex of cuts in a Stone space

TL;DR

This paper introduces a complex of cuts on Stone spaces and proves that, under certain conditions, the automorphism groups of the algebra and the space are the full automorphism group of this complex, deepening understanding of their symmetry structures.

Contribution

It defines a new complex of cuts on Stone spaces and establishes conditions under which the automorphism groups are fully characterized by this complex.

Findings

Automorphism groups coincide with the full automorphism group of the complex of cuts.

Results hold for countable Boolean algebras and spaces with at least five points.

Provides a new perspective on symmetries in Stone duality.

Abstract

Stone's representation theorem asserts a duality between Boolean algebras on the one hand and Stone space, which are compact, Hausdorff, and totally disconnected, on the other. This duality implies a natural isomorphism between the homeomorphism group of the space and the automorphism group of the algebra. We introduce a complex of cuts on which these groups act, and prove that when the algebra is countable and the space has at least five points, that these groups are the full automorphism group of the complex.