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Real spectrum versus ℓ-spectrum via Brumfiel spectrum

Abstract : It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We prove the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. The commutative unital rings and Abelian lattice-ordered groups in (2), (3), (4) all have cardinality ℵ 1. Moreover, (3) solves a problem by Mellor and Tressl.
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https://hal.archives-ouvertes.fr/hal-01550450
Contributor : Friedrich Wehrung <>
Submitted on : Friday, September 4, 2020 - 12:49:17 PM
Last modification on : Thursday, September 10, 2020 - 7:48:56 PM

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  • HAL Id : hal-01550450, version 3
  • ARXIV : 1706.09802

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Friedrich Wehrung. Real spectrum versus ℓ-spectrum via Brumfiel spectrum. 2020. ⟨hal-01550450v3⟩

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