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Article Dans Une Revue Algebras and Representation Theory Année : 2023

Real spectrum versus ℓ-spectrum via Brumfiel spectrum

Real Spectrum Versus ℓ-Spectrum via Brumfiel Spectrum

Résumé

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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Dates et versions

hal-01550450 , version 1 (29-06-2017)
hal-01550450 , version 2 (19-07-2017)
hal-01550450 , version 3 (04-09-2020)

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Citer

Friedrich Wehrung. Real spectrum versus ℓ-spectrum via Brumfiel spectrum. Algebras and Representation Theory, 2023, 26, pp.137--158. ⟨10.1007/s10468-021-10088-0⟩. ⟨hal-01550450v3⟩
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