Spectral spaces of countable abelian lattice-ordered groups

Abstract : A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian ℓ-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian ℓ-group (we say that D is ℓ-representable) iff for all a, b ∈ D there are x, y ∈ D such that a ∨ b = a ∨ y = b ∨ x and x ∧ y = 0. On the other hand, we construct a non-ℓ-representable bounded distributive lattice, of cardinality ℵ 1 , with an ℓ-representable countable L∞,ω-elementary sublattice. In particular, there is no characterization, of the class of all ℓ-representable distributive lattices, in arbitrary cardinality, by any class of L∞,ω sentences.
Type de document :
Pré-publication, Document de travail
Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee). In Corol.. 2017
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https://hal.archives-ouvertes.fr/hal-01431444
Contributeur : Friedrich Wehrung <>
Soumis le : lundi 13 février 2017 - 15:05:58
Dernière modification le : mardi 5 février 2019 - 12:12:04
Document(s) archivé(s) le : dimanche 14 mai 2017 - 14:54:58

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  • HAL Id : hal-01431444, version 2
  • ARXIV : 1701.03494

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Friedrich Wehrung. Spectral spaces of countable abelian lattice-ordered groups. Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee). In Corol.. 2017. 〈hal-01431444v2〉

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