Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

TL;DR

This paper develops new sheaf representations of MV-algebras using duality and canonical extensions, providing a unified, purely duality-theoretic approach that simplifies proofs and reveals structural insights.

Contribution

It introduces a duality-based framework for sheaf representations of MV-algebras, unifying existing results and enabling potential generalizations with minimal algebraic assumptions.

Findings

Sheaf representations of MV-algebras derived from dual space decompositions.

MV-algebras with isomorphic lattices have homeomorphic maximal spectra.

Unified duality-theoretic approach simplifies proofs and reveals structural relations.

Abstract

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.