Van Gool, Samuel Jacob; Gehrke, Mai; Marra, Vincenzo
(2014).
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Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality.
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Journal of algebra, 417, pp. 290-332.
Elsevier
10.1016/j.jalgebra.2014.06.031

Text
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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.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |

## UniBE Contributor: |
Van Gool, Samuel Jacob |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
0021-8693 |

## Publisher: |
Elsevier |

## Language: |
English |

## Submitter: |
George Metcalfe |

## Date Deposited: |
18 Mar 2015 16:31 |

## Last Modified: |
10 Sep 2015 10:42 |

## Publisher DOI: |
10.1016/j.jalgebra.2014.06.031 |

## BORIS DOI: |
10.7892/boris.65220 |

## URI: |
https://boris.unibe.ch/id/eprint/65220 |