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Cevian properties in ideal lattices of Abelian ℓ-groups

  • Miroslav Ploščica ORCID logo EMAIL logo
Published/Copyright: October 26, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 6

Abstract

We consider the problem of describing the lattices of compact ℓ-ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ-ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ1 admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ2, without a Cevian operation.

Keywords: Lattice-ordered group; ideal; algebraic lattice; spectral space
MSC 2010: 06F20; 08A30; 06E15

Communicated by Manfred Droste

Funding statement: Supported by Slovak VEGA Grant 1/0097/18.

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Received: 2021-04-01
Revised: 2021-09-17
Published Online: 2021-10-26
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
https://doi.org/10.1515/forum-2021-0074
Keywords for this article
Lattice-ordered group; ideal; algebraic lattice; spectral space

Articles in the same Issue

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
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