Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality
Chang's MV-algebras provide an algebraic semantics for Łukasiewicz infinite-valued propositional logic. As proved by the present reviewer in his paper [J. Funct. Anal. 65, 15--63 (1986; Zbl 0597.46059)], MV-algebras are categorically equivalent to abelian lattice ordered groups equipped with a distinguished (strong order) unit. As shown by a wealth of recent literature, MV-algebras go beyond the realm of nonclassical logic and universal algebra. Among others, they are deeply connected with rational polyhedra, multisets, \(C^*\)-algebras, Carathéodory and de Finetti probability theory. For a comprehensive account see the present reviewer's book [Advanced Łukasiewicz calculus and MV-algebras. Berlin: Springer (2011; Zbl 1235.03002)].NEWLINENEWLINE The aim of the paper under review is to investigate duals à la Stone-Priestley of MV-algebras: these duals are understood as lattices with extra structure naturally arising from the MV-algebraic structure. The main tool used throughout is given by ``canonical extensions''. Using the dual structure of any MV-algebra \(A\), two different representations are obtained, respectively, on the prime ideal and on the maximal ideal space of \(A\). As a main point of distinction of these representations from representations already existing in the literature, the authors give the following variant of a theorem of Kaplansky concerning the lattice of all real-valued functions on a compact Hausdorff space: If two MV-algebras have isomorphic underlying lattices then their maximal spectral spaces are homeomorphic. As noted by the authors, a very special case of this theorem holds for hyper-Archimedean MV-algebras, i.e., those MV-algebras where every prime ideal is maximal. For a related result for linearly MV-algebras with Hausdorff prime spectrum see the paper by \textit{G. A. Elliott}, \textit{R. Cignoli} and the present reviewer [Adv. Math. 101, No. 2, 166--179 (1993; Zbl 0823.46053)].
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