Representations of MV-algebras by sheaves

Ferraioli, Anna Rita; Lettieri, A.

doi:10.1002/malq.200910116

Cited by 11 publications

(11 citation statements)

References 23 publications

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“…. , n.Proof This was originally proved for MV-algebras in[21, Theorem 2.6]. In the light of Lemma 2.15 the results holds also for RMV-algebras.Theorem 2.29 (Marchioni [30, Theorem 3.5]) Every linearly ordered RMV-algebra A can be embedded in a suitable ultrapower of the RMV-algebra [0, 1] that depends only on the cardinality of A.…”

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confidence: 61%

Sheaf representations and locality of Riesz spaces with order unit

2021

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We present an algebraic study of Riesz spaces (=real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of "local" Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV-algebras. We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval [0,1]. Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces.

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confidence: 61%

Sheaf representations and locality of Riesz spaces with order unit

2021

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“…Proof. Note that f is continuous by Lemma 8.8, the fact that (19) commutes, and the definition of the quotient topology. Since q : (X, τ ↓ ) → (X/W, σ) is continuous and onto by definition, by [67,Thm.…”

Section: Direct Inspection Shows That the Underlying Distributive Latmentioning

confidence: 99%

“…Let σ denote the quotient topology that X/W inherits from (X, τ ↓ ). Then the unique function f : (X/W, σ) → (Z, τ ↓ ) making (19) commute is a homeomorphism.…”

Section: Kaplansky's Theorem For Mv-algebras: the Map Mmentioning

confidence: 99%

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

2014

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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 MValgebras 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 MValgebras, 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.

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“…They find an adjunction between MV-algebras and another version of MV-spaces. In [11] it is proposed a third kind of representation for MV-algebras using fibers that are certain local MV-algebras.…”

Section: Introductionmentioning

confidence: 99%

A representation theorem for integral rigs and its applications to residuated lattices

Castiglioni

Menni

Botero

2016

Journal of Pure and Applied Algebra

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We prove that every integral rig in Set is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.

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Representations of MV-algebras by sheaves

Cited by 11 publications

References 23 publications

Sheaf representations and locality of Riesz spaces with order unit

Sheaf representations and locality of Riesz spaces with order unit

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

A representation theorem for integral rigs and its applications to residuated lattices

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