Spectral spaces of countable abelian lattice-ordered groups

Abstract : A compact topological space X is spectral if it is sober and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. It is well known that the spectrum of an Abelian ℓ-group with unit — equivalently, of an MV-algebra — is spectral. Theorem. A topological space X is isomorphic 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 elementary sublattice. In particular, there is no first-order characterization, of ℓ-representable distributive lattices, in arbitrary cardinality.
Type de document :
Pré-publication, Document de travail
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Contributeur : Friedrich Wehrung <>
Soumis le : mardi 10 janvier 2017 - 19:32:38
Dernière modification le : mardi 5 février 2019 - 12:12:04
Document(s) archivé(s) le : mardi 11 avril 2017 - 17:15:19


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  • HAL Id : hal-01431444, version 1
  • ARXIV : 1701.03494


Friedrich Wehrung. Spectral spaces of countable abelian lattice-ordered groups. 2017. ⟨hal-01431444v1⟩



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