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.
Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality / M. Gehrke, S.J. van Gool, V. Marra. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 417:(2014 Nov 01), pp. 290-332. [10.1016/j.jalgebra.2014.06.031]
Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality
V. MarraUltimo
2014
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.Pubblicazioni consigliate
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