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

JOURNAL OF ALGEBRA

卷:417

Article

Gehrke, Mai^1,2  van Gool, Samuel J.³  Marra, Vincenzo⁴ 

[1] Univ Paris 07, CNRS, LIAFA, F-75205 Paris 13, France

[2] Univ Paris 07, Univ Paris Diderot, F-75205 Paris 13, France

[3] Univ Bern, Math Inst, CH-3012 Bern, Switzerland

[4] Univ Milan, Dipartimento Matemat Federigo Enr, I-20133 Milan, Italy

关键词: MV-algebra;    Lattice-ordered abelian group;    Distributive lattice;    Stone duality;    Priestley duality;    Sheaf representation;    Etale decomposition;    Spectral space;    Compact Hausdorff space;    Stably compact space;   

DOI  :  10.1016/j.jalgebra.2014.06.031

来源: Elsevier

【 摘 要 】

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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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